Yes it is.
Yes, it is AFAIK.
Orenthal
Clerow (Wilson)
Lemonjello and Oranjello
Yep, Uma was born Uma Karuna Thurman.
Tori Amos was born Myra Ellen Amos.
I love these names you guys are coming up with. Thanks!
All these posts and no nitpicking of the title yet? What’s the difference between “unique” and “completely unique”? Can something be “partially unique”? 
The official definition of “unique” is completely alone in the universe. Colloquially, though, it’s a synonym for “rare,” unless you’re a [del]prick[/del] traditionalist.
Would Wentworth Miller qualify? I’ve never heard of any other Wentworths.
Blossom Dearie. Eagle Eye Cherry. Kaki King.
I’ve never heard of another Tegan, so I suppose Tegan Rain Quin of Tegan & Sara would count, unless someone comes in with evidence to the contrary.
Is Wyclef Jean the name he was born with?
I can’t read Wikipedia, apparently–his name is Nelust Wyclef Jean.
And, to make a triple post because I really can’t read, according to this from the OP:
[quote]
singers and actors with unique first names (to their profession, that is)*
Sufjan Stevens.
You mentioned Loudon Wainwright III and didn’t mention his son Rufus? How many Rufus’s are there?
Rufus Thomas. I wouldn’t be surprised if that’s who Rufus Wainwright was named for. There was also, of course, a whole band called Rufus.
There have been any number of C&W singers with names that are at least unusual if not unique:
Faron Young
Ferlin Husky
Webb Pierce
Porter Waggoner
Henson Cargill
Tompall Glaser
I suggest looking up artist names in the comprehensive All Music Guide database to check for uniqueness.
Well, I was gonna say “sorta unique” but uh… Ok, so I talk funny. I’m one of those people who annoys other people by my speech habits, because I often speak and write in redundancy and cliches. It’s a bad, bad habit. 
Hey, I’ll take that. I took Pier Angeli, and besides, to not take a Nelust Wyclef would wrong on so many levels. What a name!
I don’t know any other Wentworths, in music at least.