When my brother showed photos to his young son, he'd say "...and that's me" when pointing to himself in the photo, so my nephew assumed that was his name. ("Me, Me, what's for dinner, Me?") He now understands pronouns, but seems to have also kept "Me" as his dad's name.

People don't realize that little kids talk, all the time, and especially to anyone who spends every day with them, it's clear when they are developing new concepts in their mind - they go through a cycle of linguistic scientific experimentation, testing the waters, so to speak, on their newfound grammatical mechanics or vocabulary.