For me, my organization of sounds was visually very black and white as it relates to the keys (not having any relation to whether I saw colors in my mind), and mathematical (without knowing that), and based on knowing simple tunes in various places on the piano. I didn't know about 2nds and 3rds, but I knew that there were various places I could play "Mary Had A Little Lamb" - you see? I started seeing keys as collections of sound that made music, or as shapes between two or more pitches. And, that is what my concept of the piano largely was. I am *just* brinking on understanding more about that, but it takes me years of growing formally in particular concepts, and amassing ideas as I progress, sometimes having bigger clicks sometimes smaller clicks, following inklings, pondering, and then for me to finally arrive at a point where I can see all of those things coming together to include my childhood. I have an idea that my being able to transpose something like Mary Had a Little Lamb on the piano, is something similar in concept to a child in Russia needing transpose little Czerny or Hannon exercises.
I have been working a lot with ideas on short term memory as well as long term, and this is related to how I thought as a child, too. I started developing a sort of bank in my mind and ear of sound/shape samples. But, instead of it being formal like I, IV, V, and some melody consisting in my mind as formal, numeric intervals, I thought things like "Oh, this group of notes sounds like this part of a such and such a song" and, I remember working with pitch memory A LOT, in the short term. When I played by ear I *had* to work with pitch memory as much as possible - but what exactly I would do and how I thought back then is still a bit too complex for me to take out and look at objectively with words and formal explanations.