Serena is simply describing what any of us likely experience. Case in point, I'm a really nice, submissive guy most of the time, with a long fuse, but I have a wicked temper. If I named my temper Joey, I could say, "Well, not having a great day today, Joey showed up earlier and said some things I wish he didn't say." It may sound crazier to some people if I put it that way, but it's simply another way of phrasing something mundane.
As I indicated above, naming your "personalities" can be part of the process of autotherapy, getting to know yourself and learning how to manage your moods, and it's not a bad idea unless one starts using the named personalities to deny personal culpability for their actions. Serena seems to be indicating that she doesn't "let out" certain inner voices very often, which means she is (at least somewhat) taking responsibility for what she allows herself to do and say. That being said, she could use a bit more maturity, though her outbursts are not unprecedented in the tennis world (ahem, MacEnroe).
Then again, I stopped believing in things that go bump in the night when I stopped believing in the JW religion, so I don't believe that possession is a real concept.