Does she have a sense of "self"? Meaning, does she use "mine, me, or my" when referring to herself? She's probably going through a phase where her favorite characters use 3rd person and she hears everyone else call her by her name, so she thinks that's what she should call herself, too.
You could play some turn-taking games with 3 or more people where you ask, "Whose turn is it" and whichever person has the turn says "MY TURN!" dramatically. Then they describe what they are going to do by starting with "I (want, need, have, etc.)".
Sample:
Mommy: Who's turn is it?
Daddy: MY TURN!
Daddy: I want the green one!
Mommy: OK!
Mommy: Who's next? Whose turn is it? (While blatantly staring at child)
Child target response: Me! or My turn!
Mommy: Ok! Here you go! What do you want?
Child target response: I want.......
Hopefully you get the picture. You can do it with pretty much any toy as long as you have 3 or more people. You need a "speaker" and a "model turn-taker" and the child. You can do it with ball, coloring, blocks, farm, looking at pictures in books, whatever.