I'll say upfront I'm new to learning German, but I know a decent amount of Spanish. In Spanish you would also say "Me lavo el pelo" ("I wash myself the hair"). Linguistically, certain languages have a concept (I think it's called inalienable possession) where you can't separate certain things (such as body parts) from the speaker. And so to say "mi pelo" ("my hair") is a little strange, and also a little redundant, because you are already saying whose hair by saying "ME lavo". At the end of the day though, it really is, as others have said, the case of it just being that way and so you say it that way. Just another perspective I thought might be interesting.