Is empathy discussed anywhere in the suttas?

“There is the case where a disciple of the noble ones reflects thus: ‘I love life & don’t love death. I love happiness & abhor pain. Now if I — loving life & not loving death, loving happiness & abhorring pain — were to be killed, that would be displeasing & disagreeable to me. And if I were to kill another who loves life & doesn’t love death, who loves happiness & abhors pain, that would be displeasing & disagreeable to the other. What is displeasing & disagreeable to me is displeasing & disagreeable to others. How can I inflict on others what is displeasing & disagreeable to me?’ Reflecting in this way, he himself refrains from taking life, he gets others to refrain from taking life, and he speaks in praise of refraining from taking life. In this way, his bodily behavior is pure in three ways.

“Further, he reflects thus: ‘If someone, by way of theft, were to take from me what I haven’t given, that would be displeasing & disagreeable to me. And if I, by way of theft, were to take from another what he/she hadn’t given, that would be displeasing & disagreeable to the other. What is displeasing & disagreeable to me is displeasing & disagreeable to others. How can I inflict on others what is displeasing & disagreeable to me?’ Reflecting in this way, he himself refrains from taking, by way of theft, what hasn’t been given, he gets others to refrain from taking, by way of theft, what hasn’t been given, and he speaks in praise of refraining from taking, by way of theft, what hasn’t been given. In this way, his bodily behavior is pure in three ways.

“Further, he reflects thus: ‘If someone were to commit adultery with my wives, that would be displeasing & disagreeable to me. And if I were to commit adultery with the wives of another, that would be displeasing & disagreeable to the other. What is displeasing & disagreeable to me is displeasing & disagreeable to others. How can I inflict on others what is displeasing & disagreeable to me?’ Reflecting in this way, he himself refrains from sexual misconduct, he gets others to refrain from sexual misconduct, and he speaks in praise of refraining from sexual misconduct. In this way, his bodily behavior is pure in three ways.

“Further, he reflects thus: ‘If someone were to damage my well-being by telling a lie, that would be displeasing & disagreeable to me. And if I were to damage the well-being of another by telling a lie, that would be displeasing & disagreeable to the other. What is displeasing & disagreeable to me is displeasing & disagreeable to others. How can I inflict on others what is displeasing & disagreeable to me?’ Reflecting in this way, he himself refrains from telling lies, he gets others to refrain from telling lies, and he speaks in praise of refraining from telling lies. In this way, his verbal behavior is pure in three ways.

“Further, he reflects thus: ‘If someone were to divide me from my friends with divisive speech, that would be displeasing & disagreeable to me. And if I were to divide another from his/her friends with divisive speech, that would be displeasing & disagreeable to the other. What is displeasing & disagreeable to me is displeasing & disagreeable to others. How can I inflict on others what is displeasing & disagreeable to me?’ Reflecting in this way, he himself refrains from divisive speech, he gets others to refrain from divisive speech, and he speaks in praise of refraining from divisive speech. In this way, his verbal behavior is pure in three ways.

“Further, he reflects thus: ‘If someone were to address me with harsh speech, that would be displeasing & disagreeable to me. And if I were to address another with harsh speech, that would be displeasing & disagreeable to the other. What is displeasing & disagreeable to me is displeasing & disagreeable to others. How can I inflict on others what is displeasing & disagreeable to me?’ Reflecting in this way, he himself refrains from harsh speech, he gets others to refrain from harsh speech, and he speaks in praise of refraining from harsh speech. In this way, his verbal behavior is pure in three ways.

“Further, he reflects thus: ‘If someone were to address me with idle chatter, that would be displeasing & disagreeable to me. And if I were to address another with idle chatter, that would be displeasing & disagreeable to the other. What is displeasing & disagreeable to me is displeasing & disagreeable to others. How can I inflict on others what is displeasing & disagreeable to me?’ Reflecting in this way, he refrains from idle chatter, gets others to refrain from idle chatter, and speaks in praise of refraining from idle chatter. In this way, his verbal behavior is pure in three ways."

~ the Buddha, Veḷudvāreyya Sutta: The People of Bamboo Gate, transl. Thanissaro https://www.dhammatalks.org/suttas/SN/SN55_7.html

I'm not sure if it is discussed in the suttas. From what I currently observe, there is a kind of empathy that exists as a primitive social function, in which one perceives and thus feels the experience of another; one imagines the world of another and attempts to feel what they may be feeling as a way to share the collective human conscious.

Those people who are good at doing this have what I call an exceptional degree of emotional intelligence, for being good at this means you don't allow the perception of others - and what you feel from those perceptions - to drown your objectivity.

Those who remain in the primitive functions, empathy can cause all sorts of issues including mental health problems, simply because one invests too much of their thoughts and feelings in what they perceive of others.

Principally, empathy is a primitive social function that once served the survival of the human, but its function is now caught up in thick and dense neurosis. Even teachers are guilty of this, asserting that: "you must give your thoughts to this group of suffering people. What they are going through is hideous. You need to suffer with them!". Rubbish!

Every emotional investment we make in another person, we invite the potential for pain to arise. That is the price we pay for giving ourselves in this way to another person. In doing this, all of our autonomous energy is absorbed by the relationship and our individuality disappears down the messy, unkempt toilet of behavioural dysfunctions and passive emotional coercions. Strangely, we find that we need these dysfunctions to feel whole, noticed and worthy; its the global culture's way of feeling connected because society teaches that one must 'feel' our fellow's pain, and if you don't, you're seen as cold and heartless! What a terribly silly philosophy!

Now, you asked about a skilful way to utilize empathy: firstly, one must claim back their autonomy. Secondly, one must then see what is actually happening in the world before setting back in. Thirdly, having seen the state of affairs, and having access to the bigger picture, one finds they no longer have to do anything, for everything is doing itself. Yes, there can be listening and feeling, but it is much more efficient because there is nobody there thinking that they are part of the show. The show of appearances appears in a non-relational and non-experiential none-event called the absolute!

Ven. Sujato uses "empathy" once in his translations here:

SN10.2:4.1: If you instruct others
SN10.2:4.2: with a mind clear and confident,
SN10.2:4.3: your compassion and empathy
SN10.2:4.4: don’t create attachments.”

In general however, we have this about understanding the minds of othes:

DN2:91.2: They understand the minds of other beings and individuals, having comprehended them with their own mind.

DN2 continues in detail:

DN2:91.3: They understand mind with greed as ‘mind with greed’,
DN2:91.4: and mind without greed as ‘mind without greed’.
DN2:91.5: They understand mind with hate …
DN2:91.6: mind without hate …
DN2:91.7: mind with delusion …
DN2:91.8: mind without delusion …

It takes time to understand our own minds. It takes time to understand and let go of defilements so that true compassion and empathy can guide our thoughts, speech and actions.