What bands most "owned" their sound?

Primus.

Chicago.
We left half way though their 4th of July show because all their songs sound the same.

Depeche Mode.

E.

Oingo Boingo.

The Band. The Danko/Helm tempo is pretty unmistakeable.

The Beach Boys rise to close the top. First, they invented the surf genre, more or less, but no other surf band sounded like anything Brian and the Boys together. He -They - had a nothing like it sound that is instantly recognizable and completely unmatched.

Sonic Youth, Tool, Nine Inch Nails, The Doors, Korn, Linkin Park

New Order

Only one band ever sounded like Motorhead.

Santana was the first Band to come to my mind

It’s not all rock and roll. The Crusaders(aka The Jazz Crusaders) have such a distinctivesound that you can hear it in the first few notes.

Great pick. Has there been another group as popular as them ever had virtually no one try to sound like them?
I’ll add these artists from the edge:
Captain Beefheart and the Magic Band. The Captain’s vocals are actually probably the least distinctive part of the band.
The Shaggs, who nobody could copy because they had no idea what they were doing.

I’ll nominate ELP. They had a very distinct sound and style.

I think it’s questionable whether The Shaggs should even be considered since their albums are the musical equivalent of vanity press books, but if they should be considered, then they may very well be the winners of this category.
The Shaggs are horrible, but somehow I find them strangely compelling and every now and then I get on a Shaggs kick and play them on Youtube. My Pal Foot Foot and Philosophy of the World are the ones I always remember.

Don’t forget Iron Butterfly. Am I the only one who played the flip side? Or at least heard their other albums? Very distinctive sweet organ and mega sustain guitar.

Fairport Convention had their moments too. Especially when Sandy Denny was singing with them.

There was The Byrds too, at least until Crosby left.

Traffic?

T Rex.

Very distinctive sound that nobody has ever reproduced. And every T Rex cover I’ve ever heard is terrible, with the exception of Bauhaus’ cover of Telegram Sam, which they completely rearranged.

Boston and Steely Dan, both recognizable immediately.

Meat Loaf/Jim Steinmann.

I once heard a song on the radio for the first time and thought “Sounds like a Meat Loaf song, but the voice isn’t his. I bet it’s Jim Steinmann.” I was right.

I’m not sure whether I agree with this. Both bands explored a lot of musical territory over the course of their respective careers. Both bands established a distinctive early style but then branched out from there. And I can think of more songs by other artists that, deliberately or not, sound like the Beatles than that sound like the Kinks. Though I may be misled here by the number of songs that are reminiscent of particular songs by the Beatles.
As for the general question: A lot of bands have developed a distinctive sound of their own. That’s almost a requirement for being a great band. It’s hard to single out one or a few that most “owned” their sound. Though CCR isn’t a bad example at all.

Status Quo

Dread Zeppelin

U2, obviously; nobody really plays guitar like The Edge, at least not in any band I’ve heard, and when you add Bono’s distinctive voice, their music is instantly identifiable, whether it’s a record made in the early 80s of a record made three years ago.

Our beloved Tragically Hip, soon to play their last concert, play music best described as Tragically Hip music; it’s kinda bar rock and kinda blues and kinda hard rock but it’s not quite any of those things, it’s Tragically Hip Music. It is inconceivable to imagine U2 successfully pulling off “New Orleans Is Sinking,” and equally inconceivable to imagine the Hip successfully executing “Pride.”

And if anyone else ever sounded like Rage Against The Machine, I do not remember who it was.