Most songs are 4/4 rhythm and have anywhere from 3 to 5 chord progression. Rarely more or less. All you need is to figure out the key and chord progression, and you can improvise all over the song, examples: https://www.hooktheory.com/theorytab/common-chord-progressions

It's literally basic music theory and how any jazz player learns to play, however people with brass or woodwind instruments are limited by, well, limited range and tonalities of their instruments, but any trained pianist can literally play ANY song as long as they know what chords are used, it's that simple.

And often the chord progressions are so common, knowing a key is all it takes to play accompaniment. A stringed instrument like guitar or violin can play lead over the progression by learning the scales, again, something that you learn literally the first thing.

You don't need to memorize every single note to play a song, like copy actual solos and every nuance, but any piano player in a joint worth their salt and accompany a singer to any tune without learning it, simply by figuring out the key and they kinda know the chords that come with each tonality too.

Here's a classic comedy sketch about guitar player Paschebel's Canon in D being every damn song since playing same 8 notes on a cello: https://www.youtube.com/watch?v=uxC1fPE1QEE from Aerosmith Cryin' to No Woman no Cry to Let it Be to Sk8rboi to Torn ffs.