How rare is it for someone to hear a piece of music, say semi-complex classical piece, once for the first time and be able to play it from memory at once?

I remember a scene like this in Amadeus. But is this somewhat common for gifted musicians? Or rare?

I was watching a show and this pre-teen kid listened to a pianist composer play an original work once and then was able to play it immediately. The composer even admitted he made a mistake in playing his own piece and the kid even copied the mistake.

Just another gifted kid? It another Mozart in the making?

I'm about as far from a genius musician as you can get, so take this with a grain of salt. Once you understand the basic structure of music, it becomes surprisingly easy to memorize a specific piece of music (with lyrics being an exception if there are any). You kind of say to yourself, "Oh, that's where the composer is going. Yeah, an AABA song structure built on a 1-4-5 chord progression." Like that.

That is my favorite scene from Amadeus.

I believe what you describe is possible, yet the source of the talent transcends the typical meaning of genius. I have a violinist friend who can play any song after hearing it, yet could not do the same with topics like physics or math.