John Wayne took his last name from Revolutionary War General “Mad Anthony” Wayne. It actually was supposed to be Anthony Wayne, but the studio rejected it.
Winston S. Churchill, author and British Prime Minister, shared a name with the forgotten American author Winston Churchill.
Chevy Chase
Franz Ferdinand the band, named afterFranz Ferdinand, the archduke.
The Marx Brothers were born the Marx Brothers. They didn’t take on the name. And none of them was named Karl. Adolph, yes. Karl, no.
A slight hijack from the OP, but goes along with the above post. Charlton Heston’s born name was John Carter. Not that anyone was using it, but I imagine the Burrough’s connotations would have been difficult. However, he’d been using the name Heston since high school(his step-father’s name) and used his mother’s maiden name Charlton after that.
I can imagine him arguing with obnoxious fans who didn’t believe him: “My name really is Paul Revere, Dick!”
Iwas going to say Alice Cooper but apparently that was just a story the band made up.
Wait. What? That’s the sort of name you run away from, not to!
Davy Jones. It’s his real name, but he shares it with the maritime legend.
Little Nell, from Rocky Horror (Has Tiny Tim been mentioned?)
Marc Anthony
I know. The OP asked for people who either took their names from famous people or people who shared names with famous people - being born with them, like all the others I suggested.
That was my first thought too.
Sebastian Bach from Skid Row
Nicholas Cage allegedly named himself after superhero Luke Cage
Actress Holly Golightly
I assumed that sharing them had to mean the whole name. Otherwise you descend into absurdity like Dinah Washington and Aretha Franklin. Or Count Basie. 
Fannie Flagg’s real name is Patricia Neal (Oscar winning star/Mrs. Roald Dahl much better known in the '50s when Fannie was starting out).
Chase was apparently nicknamed after a song (“The Ballad of Chevy Chase”), which in turn refers to a Scots-English battleground. I thought he named himself from the town in Maryland, but he grew up in New York so it’s unlikely. Either way he was named after a place, which is a category we haven’t yet considered.
In the mid-70s the Australian pop music scene was graced by a truly dreadful pop singer who called himself - or more likely his agent made him call himself - William Shakespeare.
My name is Bond. James Bond, ornithologist. I chase birds.
So who did Yahoo Serious name himself after?
We had the novelty pop singer Napoleon XIV, named after a French king and emperor, but I suppose that’s too ridiculous to count.
And then we had the pop singer Phoebe Snow, named after an old-time railroad advertising icon, about whom Cecil once wrote a snippy column.
John Adams is both a contemporary opera composer and the 2nd president of the USA.
Justus E McQueen is better known as the obscure actor L.Q. Jones. His most recent role was Chuck Akers, the country/gospel singer who dies in A Prairie Home Companion.
His first part was as Pvt. L. Q. Jones in the 1955 movie Battle Cry He changed his name after that part.
He has never been a headliner, but he has 158 credited performances listed at IMDB.