In Buddhism, yes.

For Hindus, well, it's complicated.

For other people who happen to believe in reincarnation?
That would be anybodys guess, I guess.

I think it depends on what religion we’re talking about. OP didn’t specify one so, in general, if there was an actual process that put “you” into another body, assuming it’s human, then would it be random or “good?” Theoretically, life is what you make of it, so no “good” is really guaranteed if you mess it up. But starting out with more options than another person would give you an advantage.

I think if it was real, it would be random. Like, you die and you pop into whatever baby was next in line and that’s the gamble.

Of course, none of that is real. There is no god, no afterlife, nothing but atoms in an amazingly random configuration to produce “life,” as we know it.

I have also heard buddhist view of reincarnation as simply the act of identifying with yourself in each new moment. Since enlightenment involves disillusionment with self, this is how enlightenment stops the process of reincarnation.

I think this is correct. Had a friend ages ago who was Buddhist and I remember her saying something like the best thing to be reincarnated as was a butterfly because their lives are so short and that reincarnation was undesirable because enlightenment was the real goal. If you attained enlightenment you won the game, so to speak.