As a spinoff thread from the recent post about memorization possibly affecting how musically one plays, and in recognition of the fact that I haven't created a thread over here in months (I think!), I decided I might share some of my personal observations.
When I started playing the piano, I was convinced that the best way to play musically was to play blind. This was probably because it seemed like that would allow the ear the most room to actively listen to the music being produced. I was really inspired by the _Mental Play concept described in C Chang's Fundamentals of Piano Practice, which was basically audiation that was presented in a way in which it seemed as if it was an ability most of us have naturally.
I remember feeling that if only you could visualize the sound well enough that you want to produce, your fingers and hands would magically comply if the intent was signaled strongly enough.
Usually, if a piece of music or a passage moved me, it would be memorized really quickly and seemingly without much effort. It was not "automatic", but it was fairly rapid, over the course of a few days or weeks. In that sense, I was probably a "natural memorizer". The stronger the emotional connection, the easier it would be for me to memorize and internalize music. And in fact, in almost all cases of people that I know of surprising their teachers by learning a piece of music incredibly quickly or making crazy progress, it was spurred on by a specific passion and emotional connection for that piece that bordered on obsessive or even fanatical.
The medium using which I would learn (by ear, synthesia, copying hand positions, sheet music) didn't seem to matter as much; as long as I got the notes and had a specific interest for the piece, memorization would ensue fairly automatically. There would be fuzziness of the memory in places, but the general gist and most of the information would be somewhat passively absorbed. This is still mostly the case, where I tend to not end up "read-playing" as much as I probably should because things become quasi-memorized after a few repetitions and kind of enter short-term memory. There is a point where the information is "in", so to speak, after which it has made an "impression" that may or may not end up being truly remembered, but before which it hasn't made any imprint on the mind. Sometimes, stopping around that point is enough for something to become memorized long term, but stopping before then will basically never create a lasting impact.
Memorization for me has also been very closely tied to playing by ear and aural imagination, as well as visual/sensory imagination -- in a way akin to having a very vivid imagination/hyperphantasia. If I was really "tuned in" to a piece of music and if I could play it while closing my eyes, I would sometimes start imagining a movie to go along with it, and that tied directly to the strength of the emotional connection and the memory in a feedback loop. This possibility of "layering" different modalities of experience when you play a piece memorized, I feel, is a strong part of what can make it more effective as a way to internalize music more deeply than reading from the score.
Listening experience, imo, is also a very important kind of memory. The pieces that you know well enough to the point where you can audiate them or hum them really seem to form a basis for so much. The memorized music (both piano pieces and aural memory) also forms a basis for improvisation and composition, that leads fairly naturally to music theory and a deeper understanding of pieces, and then the ensuing patterns (both generated and learned/encountered ones) make future memorization easier in a feedback loop.
Some of this can be learned, some of it is probably innate, and I'm still trying to decipher which is which.