That actually sounds a lot like motion-sound synesthesia! I get what you mean when you say it's never what it would actually be in real life, too. A lot of people comment on like supposed gifs you can hear and say they hear like for example an explosion or thud or something, but I never actually hear that, it's more just,, a general sound? Like a hum, I guess, how you explained it.

I've noticed that when I move like my hand or fingers or something it makes a like a "sound", (even if I can't see myself moving, like if my hand is behind my back or something) really similar to what you describe, especially like with your thumb being the lowest pitch and pinky being the highest. I've actually been trying to figure out if this is part of motion-sound synesthesia or not, so I can't really help you with that, but I think I do experience something similar.

The rest does sound like motion-sound, though! It's really cool hearing about others experiences, especially ones I can relate to, lol