that's also an age thing - kids need to be a bit older to make the transition from "name" to "I" (I can't remember exactly, somewhere between 2 and three I think, but could be wrong??)
My DD1 seems to have this backwards - can't tell me her name when asked, but calls herself "me" for everything. If I ask her name she says "me!", though she might just be saying the middle syllable of her name...