I've been mulling over this very question recently.
This post assumes that you're a one man team. If you're a team, you should probably come up with a team name.
I think traditional game studios are at a certain disadvantage because it's harder to feel a personal, human connection to them. There's often a guy who represents projects publicly—like a Todd Howard—but ultimately you have many, many people who work on a single game, and one man, even under a creative role, can't adequately speak for an entire team or studio. The person(s) in charge of social media typically don't work on the games anyway. You don't assume Dan Houser is running Rockstar Games' Twitter. It's just yet another inhuman social media feed that behaves more or less the same as every other one.
We're inherently interested in other humans, and indie game developers can capitalize on that interest better than any game studio can. You have a story to tell, a personality and a way of thinking that you can let shine through, not only directly in social media, articles and interviews, but in your games too. If your games are personal—which is another advantage indies have over most studios—I think it's to your benefit to identify as a yourself and to embrace that.
It seems like it would be difficult to successfully market an independent game while avoiding the public spotlight, and trying to do so would only put you at a disadvantage, so you and your name, Phil Andy (if that is your real name), will already be a factor by default. If you introduce a "studio" name, aren't you just making things more confusing?
On the other hand, a good studio/company/whatever name is usually going to be easier to remember than a real name. Most of us have rather unoriginal names, and for me personally, even if you have an original name, I still have a hard time remembering it. Studio/company/whatever names are usually made to be memorable and original.
Something I don't fully understand is why solo indie game developers come up with studio names and don't embrace them at all. Why introduce the name into the mix in the first place? A prime example of this is Jonathan Blow, the creator of Braid. He owns Number None, Inc., but rarely ever mentions it. Is it related to the nuances of business and company registration?
At the end of the day, it probably doesn't matter too much. I don't think something like this is going to keep you from being successful.
