Is it possible to exactly reverse engineer an unknown cipher inside a black box? Consider the following arrangement:-



There is an unknown cipher algorithm operating inside the black box. You can enter infinite key /plain text pairs, and observe   all of the outputs. The box might just as well be protecting a short friend/foe challenge as an entire 4K UHD movie. Any counter modes, streams, IVs, nonces or padding is handled inside the box. All you need to do is input any size of plain text. The cipher is a world class modern type with strength equivalent to *fish, Rijndael or ChaCha, but not one of those.  It may have come from outer space, or perhaps in a vision. The point is that no one has ever seen it. It might even be a totally unknown NSA Suite A algorithm.

Given unlimited input output tuples, is it possible to reverse engineer the algorithm exactly so that it could be  used reliably for future encryptions?

Note: Hypothetical unknown cipher - security in obscurity? came up, but honestly I can't understand   the question.
Judging by your comments, you are imagining a box that accepts inputs of arbitrary length. Hence it is a function $f:\{0,1\}^*\to\{0,1\}^*$. There are uncountably many such functions but only countably many programs you could propose as the "reverse engineered" result. This makes your problem fundamentally impossible.