💚What does the idea of "separation" have to do with the Buddha's teaching?

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OP: What is meant with the idea of separation in the Buddhist interpretations of the Buddha's teaching

Separation of an individual being happens when there's craving and clinging.

As he was sitting there he said to the Blessed One: "'A being,' lord. 'A being,' it's said. To what extent is one said to be 'a being'?"

"Any desire, passion, delight, or craving for form, Radha: when one is caught up there, tied up there, one is said to be 'a being.'

"Any desire, passion, delight, or craving for feeling... perception... fabrications...

"Any desire, passion, delight, or craving for consciousness, Radha: when one is caught up there, tied up there, one is said to be 'a being.'
SN 23.2

The individual being is the carrier of the burden.

When the burden is carried, then the carrier of the burden, the separate individual being, will appear.

The carrying of the burden is craving.

“And what, bhikkhus, is the burden? It should be said: the five aggregates subject to clinging. What five? The form aggregate subject to clinging, the feeling aggregate subject to clinging, the perception aggregate subject to clinging, the volitional formations aggregate subject to clinging, the consciousness aggregate subject to clinging. This is called the burden.

“And what, bhikkhus, is the carrier of the burden? It should be said: the person, this venerable one of such a name and clan. This is called the carrier of the burden.

“And what, bhikkhus, is the taking up of the burden? It is this craving that leads to renewed existence, accompanied by delight and lust, seeking delight here and there; that is, craving for sensual pleasures, craving for existence, craving for extermination. This is called the taking up of the burden.

“And what, bhikkhus, is the laying down of the burden? It is the remainderless fading away and cessation of that same craving, the giving up and relinquishing of it, freedom from it, non-reliance on it. This is called the laying down of the burden.”
SN 23.22

And craving (tanha) is a habit of reification or objectification-classification (papanca), which separates into objects of self and non-self.

"I ask the kinsman of the Sun, the great seer, about seclusion & the state of peace. Seeing in what way is a monk unbound, clinging to nothing in the world?" "He should put an entire stop to the root of objectification-classifications: 'I am the thinker.'
Snp 4.14

Commentary on Snp 4.14:
On objectification-classifications and their role in leading to conflict, see Sn 4.11 and the introduction to MN 18. The perception, "I am the thinker" lies at the root of these classifications in that it reads into the immediate present a set of distinctions — I/not-I; being/not-being; thinker/thought; identity/non-identity — that then can proliferate into mental and physical conflict. The conceit inherent in this perception thus forms a fetter on the mind. To become unbound, one must learn to examine these distinctions — which we all take for granted — to see that they are simply assumptions that are not inherent in experience, and that we would be better off to be able to drop them.

OP: What makes one person's karmic stream separate from another person's karmic stream?

New kamma comes from actions done by the separate individual being, the carrier of the burden.

However, old kamma is said to simply come from volition, and something to be felt. Who it comes from, is not mentioned.

“And what, bhikkhus, is old kamma? The eye is old kamma, to be seen as generated and fashioned by volition, as something to be felt. The ear is old kamma … The mind is old kamma, to be seen as generated and fashioned by volition, as something to be felt. This is called old kamma.

“And what, bhikkhus is new kamma? Whatever action one does now by body, speech, or mind. This is called new kamma.
SN 35.146

In fact, karmic streams are neither joined nor separate.

Instead, the Buddha spoke of dependent origination.

“How is it, Master Gotama: is suffering created by oneself?”

“Not so, Kassapa,” the Blessed One said.

“Then, Master Gotama, is suffering created by another?”

“Not so, Kassapa,” the Blessed One said.

“How is it then, Master Gotama: is suffering created both by oneself and by another?”

“Not so, Kassapa,” the Blessed One said.

“Then, Master Gotama, has suffering arisen fortuitously, being created neither by oneself nor by another?”

“Not so, Kassapa,” the Blessed One said.

“How is it then, Master Gotama: is there no suffering?”

“It is not that there is no suffering, Kassapa; there is suffering.”

“Then is it that Master Gotama does not know and see suffering?”

“It is not that I do not know and see suffering, Kassapa. I know suffering, I see suffering.”

“Whether you are asked: ‘How is it, Master Gotama: is suffering created by oneself?’ or ‘Is it created by another?’ or ‘Is it created by both?’ or ‘Is it created by neither?’ in each case you say: ‘Not so, Kassapa.’ When you are asked: ‘How is it then, Master Gotama: is there no suffering?’ you say: ‘It is not that there is no suffering, Kassapa; there is suffering.’ When asked: ‘Then is it that Master Gotama does not know and see suffering?’ you say: ‘It is not that I do not know and see suffering, Kassapa. I know suffering, I see suffering.’ Venerable sir, let the Blessed One explain suffering to me. Let the Blessed One teach me about suffering.” “Kassapa, if one thinks, ‘The one who acts is the same as the one who experiences the result,’ then one asserts with reference to one existing from the beginning: ‘Suffering is created by oneself.’ When one asserts thus, this amounts to eternalism. But, Kassapa, if one thinks, ‘The one who acts is one, the one who experiences the result is another,’ then one asserts with reference to one stricken by feeling: ‘Suffering is created by another.’ When one asserts thus, this amounts to annihilationism. Without veering towards either of these extremes, the Tathagata teaches the Dhamma by the middle: ‘With ignorance as condition, volitional formations come to be; with volitional formations as condition, consciousness…. Such is the origin of this whole mass of suffering. But with the remainderless fading away and cessation of ignorance comes cessation of volitional formations; with the cessation of volitional formations, cessation of consciousness…. Such is the cessation of this whole mass of suffering.’”
SN 12.17

OP: In accordance with the Buddha's teaching, what can we point to and say, that is where (any) two things are separate or not separate?

You cannot really find the separation of an individual being.

“What do you think, friend Yamaka, do you regard form as the Tathagata?”—“No, friend.”—“Do you regard feeling … perception … volitional formations … consciousness as the Tathagata?”—“No, friend.”

“What do you think, friend Yamaka, do you regard the Tathagata as in form?”—“No, friend.”—“Do you regard the Tathagata as apart from form?”—“No, friend.”—“Do you regard the Tathagata as in feeling? As apart from feeling? As in perception? As apart from perception? As in volitional formations? As apart from volitional formations? As in consciousness? As apart from consciousness?”—“No, friend.”

“What do you think, friend Yamaka, do you regard form, feeling, perception, volitional formations, and consciousness taken together as the Tathagata?”—“No, friend.”

“What do you think, friend Yamaka, do you regard the Tathagata as one who is without form, without feeling, without perception, without volitional formations, without consciousness?”—“No, friend.”

“But, friend, when the Tathagata is not apprehended by you as real and actual here in this very life, is it fitting for you to declare: ‘As I understand the Dhamma taught by the Blessed One, a bhikkhu whose taints are destroyed is annihilated and perishes with the breakup of the body and does not exist after death’?”

“Formerly, friend Sāriputta, when I was ignorant, I did hold that pernicious view, but now that I have heard this Dhamma teaching of the Venerable Sāriputta I have abandoned that pernicious view and have made the breakthrough to the Dhamma.”
SN 22.85

In fact, it's hard to find the separation of anything. All separation is simply based on convention, as we can see from the chariot analogy.

“Why do you believe there’s such a thing as a
‘sentient being’?
Māra, is this your theory?
This is just a pile of conditions,
you won’t find a sentient being here.

When the parts are assembled
we use the word ‘chariot’.
So too, when the aggregates are present
‘sentient being’ is the convention we use.

But it’s only suffering that comes to be,
lasts a while, then disappears.
Naught but suffering comes to be,
naught but suffering ceases.”
SN 5.10

And convention comes from reification or objectification-classification or proliferation (papanca).

“Venerables, dependent on the existence of an eye and a visible object, eye-consciousness arises. The combination of the three is sense-contact. Because of sense-contact, there is feeling. What one feels, one identifies;2 what one identifies, one thinks about; what one thinks about, one proliferates about; what one proliferates about, with that as its source, identification and conceptualization based on proliferation beset a man in regard to visible objects cognizable by the eye in the past, present, and future. Venerables, when there is an ear and a sound… a nose and a fragrance… a tongue and a taste… a body and a tangible object… a mind and a mental object, mind-consciousness arises. The combination of the three is contact. Because of contact, there is feeling. What one feels, one identifies; what one identifies, one thinks about; what one thinks about, one proliferates about; what one proliferates about, with that as its source, identification and conceptualization based on proliferation beset a man in regard to mental objects cognizable by the mind in the past, present, and future.
MN 18

OP: what does it have to do with the 3 marks of existence?

Sabbe sankhara anicca - all conditioned and compounded things are impermanent.

Reification or objectification-classification (papanca) is the process through which conditioned and compounded things are delineated into self and non-self objects.

Craving is the habit of reification. It's the carrying of the burden.

Through craving, the individual being gets separated or delineated, and starts existing as the carrier of the burden.

Sabbe dhamma anatta - all phenomena is not self.

So, ultimately, a separate individual being and his separate karmic stream cannot be found.

Sabbe sankhara dukkha - all conditioned and compounded things are suffering.

All that exists is suffering (first noble truth), not individual beings or the karmic stream of individual beings.

And one last sutta to reiterate this message:

"Suppose there were a king or king's minister who had never heard the sound of a lute before. He might hear the sound of a lute and say, 'What, my good men, is that sound — so delightful, so tantalizing, so intoxicating, so ravishing, so enthralling?' They would say, 'That, sire, is called a lute, whose sound is so delightful, so tantalizing, so intoxicating, so ravishing, so enthralling.' Then he would say, 'Go & fetch me that lute.' They would fetch the lute and say, 'Here, sire, is the lute whose sound is so delightful, so tantalizing, so intoxicating, so ravishing, so enthralling.' He would say, 'Enough of your lute. Fetch me just the sound.' Then they would say, 'This lute, sire, is made of numerous components, a great many components. It's through the activity of numerous components that it sounds: that is, in dependence on the body, the skin, the neck, the frame, the strings, the bridge, and the appropriate human effort. Thus it is that this lute — made of numerous components, a great many components — sounds through the activity of numerous components.'

"Then the king would split the lute into ten pieces, a hundred pieces. Having split the lute into ten pieces, a hundred pieces, he would shave it to splinters. Having shaved it to splinters, he would burn it in a fire. Having burned it in a fire, he would reduce it to ashes. Having reduced it to ashes, he would winnow it before a high wind or let it be washed away by a swift-flowing stream. He would then say, 'A sorry thing, this lute — whatever a lute may be — by which people have been so thoroughly tricked & deceived.'

"In the same way, a monk investigates form, however far form may go. He investigates feeling... perception... fabrications... consciousness, however far consciousness may go. As he is investigating form... feeling... perception... fabrications... consciousness, however far consciousness may go, any thoughts of 'me' or 'mine' or 'I am' do not occur to him."
SN 35.205

ruben2020 36854

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There's a great book on the subject, The Rice Seedling Sutra. Buddha's Teachings on Dependent Arising by Yeshe Thabkhe.

While I'm not at all satisfied with its interpretation of the underlying sutra material, to which I don't think it does justice, I find its commentary useful on its own accord as a deep and accurate foray into the subject of Sankharas or compound phenomena.

If you remember, in Buddhism all dharmas except Nirvana (and perhaps some controversial ones like "space" that the schools don't agree on) are considered compound phenomena. This means they are temporary (transitory) bundles of other dharmas, which themselves are compound and so on ad infinitum.

This image is directly connected to the two of Three Marks Of Existence, because being temporary arrangements the compound dharmas are automatically 1) impermanent and 2) without svabhava or core identity.

This description of dharmas being in constant flux, being mere temporary arrangements of their constituent dharmas which themselves made up of yet other transitory dharmas on their separate ways to other configurations hopefully paints a clear picture that our idea of distinct entities is a gross simplification of a point-in-time snapshot of how things are. If you watch reality for any sufficiently long duration of time, it's much more like the clouds in the sky or the dancing flames in a burning campfire - with shapes appearing, morphing, mixing with nearby stuff, disbanding, then reappearing again in similar patterns but never quite the same. An individual cloud or an individual tongue of flame is something we can discern at a point in time but as we watch over time things are not as distinct at all.

The same can be said about people. We appear distinct and finite at the first glance but if you look carefully you'll see we are both in constant exchange with our environment and our underlying building blocks are dharmas on their ways to other destinations, only temporarily arranged to form Andriy or Bob.

And yet there's certainly some continuity within one life, and a lesser but still discernible continuity from parent to children, from teacher to students, and so on.

So if we are to be honest and accurate we must speak carefully and avoid blanket statements like "everything is one" or "all things are separate". Reality has some degree of continuity and that's why we have things like causality, cyclical tendencies, evolution, karma, and so on. Reality also has things mixing, morphing, falling apart, and continuing on their separate ways. It's a little bit of both.

What's important to understand is that "a thing" is a boundary and a label assigned conventionally, within a certain context. A thing is not something inherently delineated in reality, it's an abstraction. Therefore to speak about separate things or same thing is necessarily a question of semantics, it's always based on some definitions, some assumptions, some conventions in some context.

That's exactly what Buddha pointed out when he said that our problems can be traced back to our definitions but not beyond. Our problems only exist within the space of our definitions.

Therefore to learn to go beyond definitions, beyond boundaries, beyond the simplistic duality of separate vs one, beyond the simplistic extremes of mortality and immortality, to see the field of compound transient arrangements as the medium of both continuity/karma and discontinuity/impermanence - is the path to Freedom the Buddha had taught.