Inner strength as a great river

The term "river" is used in the Pali suttas, but usually referring to the flood of sensuality or samsara which carries people away (SN 35.200) -- or see also "River" in this brief "Index of Similes"

The closest Buddhist reference I know of, to the analogy you cited, is this more-or-less famous "Zen story":

Great Waves

In the early days of the Meiji era there lived a well-known wrestler called O-nami, Great Waves.

O-nami was immensely strong and knew the art of wrestling. In his private bouts he defeated even his teacher, but in public he was so bashful that his own pupils threw him.

O-nami felt he should go to a Zen master for help. Hakuju, a wandering teacher, was stopping in a little temple nearby, so O-nami went to see him and told him of his trouble.

"Great Waves is your name," the teacher advised, "so stay in this temple tonight. Imagine that you are those billows. You are no longer a wrestler who is afraid. You are those huge waves sweeping everything before them, swallowing all in their path. Do this and you will be the greatest wrestler in the land."

The teacher retired. O-nami sat in meditation trying to imagine himself as waves. He thought of many different things. Then gradually he turned more and more to the feeling of the waves. As the night advanced the waves became larger and larger. They swept away the flowers in their vases. Even the Buddha in the shrine was inundated. Before dawn the temple was nothing but the ebb and flow of an immense sea.

In the morning the teacher found O-nami meditating, a faint smile on his face. He patted the wrestler's shoulder. "Now nothing can disturb you," he said. "You are those waves. You will sweep everything before you."

The same day O-nami entered the wrestling contests and won. After that, no one in Japan was able to defeat him.

It's possible that if the movie's script-writer knew anything about Buddhism, they might have known this story (it was published in Zen Flesh, Zen Bones).

I'm guessing that it's Mahayana if anything.

The Pali does has doctrine about "strength" -- see Vīrya.

But the term "inner" strength puts me in mind of what may be later concepts -- like "Buddha nature" or "original face" -- of something primordial that remains if you can escape superficialities.

But one of the topics that's central to both forms of Buddhism is escaping from "obscurations" or "hindrances" -- which could include "fear" (as in the OP), or "bashfulness" (presumably a form of conceit) as in the Zen story.

I'm not sure it needs to be said, but we can't really take Kung Fu (the movie or the series) as a proper representation of Buddhist thought. At best, it's a Hollywoodized representation of generic East Asian mysticism. The river analogy is far more prominent in Daoism that Buddhism, and while Chan Buddhism did incorporate a lot of Daoist philosophy, this doesn't strike me as quite on point.

I suspect what you're hearing is '80s martial arts mysticism, which worked its way into US culture through Bruce Lee and his students more than Buddhist meditative centers. The sentiment is quite similar to things I've heard in discussions of Tai Chi, and probably extends to other 'circular' martial arts as well.

It seems like an empty sentiment at first, but on reflection there is something in this statement that makes me think it must come from some long lived idea. Is this just a screen writer's fancy or some reference to an old Buddhist idea?

While there's no 1 to 1 exact matching between what's said in a movie and the Buddha's teaching, the idea's always been there:

There are no painful mental states, chieftain, in one without longing. In one whose fetters are ended, all fears are overcome. With the ending of [craving] the guide to becoming, when phenomena are seen for what they are, then just as in the laying down of a burden, there's no fear in death.

I've lived well the holy life, well-developed the path. Death holds no fear for me. It's like the end of a disease.

I've lived well the holy life, well-developed the path, seen states of becoming as devoid of allure, like poison spit out after it's drunk.

One gone to the far shore without clinging without effluent his task completed, welcomes the ending of life, as if freed from a place of execution. Having attained the supreme Rightness, unconcerned with all the world, as if released from a burning house, he doesn't sorrow at death.

Whatever's compounded, wherever a state of becoming's obtained, all that has no one in charge: so says the Great Seer. Whoever discerns this, as taught by the Awakened One, would no more grasp hold of any state of becoming than he would a hot iron ball. I have no 'I was,' no 'I will be.' Fabrications will simply go out of existence. What's to lament there in that? For one who sees, as it actually is, the pure arising of phenomena, the pure seriality of fabrications, there's no fear. When seeing the world with discernment as on a par with grass & twigs, finding no 'mine-ness,' thinking, 'There's nothing of mine,' he feels no sorrow. Dissatisfied with this carcass, I'm unconcerned with becoming. This body will break up and there will not be another. Do as you like with this carcass. From that I will feel neither hatred nor love. ~~ Thag 16.1 ~~

💚That sounds very much like something a Buddhist master would say.

"Do not go in fear, fear is eternal darkness, but instead go with inner strength which is like a deep river into which all streams flow;"

Sounds like what happens when a person practices Satipatthana mindfulness: They look at what is coming through the streams of seeing, hearing, thinking and feeling continuously on a moment-by-moment basis as a part of the Eightfold Noble Path.

The Buddha taught about fear in several ways. Here is one:

AN9.5:5.1: A noble disciple who has these four powers has got past five fears.
AN9.5:5.2: What five?
AN9.5:5.3: Fear regarding livelihood, disrepute, feeling insecure in an assembly, death, and bad rebirth.

What has the strength to overcome those fears?

AN9.5:5.4: Then that noble disciple reflects:
AN9.5:5.5: ‘I have no fear regarding livelihood.
AN9.5:5.6: Why would I be afraid of that?
AN9.5:5.7: I have these four powers:
AN9.5:5.8: the powers of wisdom, energy, blamelessness, and inclusiveness.
AN9.5:5.9: A witless person might fear for their livelihood.
AN9.5:5.10: A lazy person might fear for their livelihood.
AN9.5:5.11: A person who does blameworthy things by way of body, speech, and mind might fear for their livelihood.
AN9.5:5.12: A person who does not include others might fear for their livelihood.
AN9.5:5.13: I have no fear of disrepute …
AN9.5:5.14: I have no fear about feeling insecure in an assembly …
AN9.5:5.15: I have no fear of death …
AN9.5:5.16: I have no fear of a bad rebirth.
AN9.5:5.17: Why would I be afraid of that?

As you can see, this may raise more questions whose answers can also be found in the suttas. May your journey into the suttas be fruitful and bring you peace.