The one that comes to mind is “Jocko Homo,” where they sing “We Are Devo!” so, yeah, that certainly counts.
“Gorillaz got the bass drum” M1A1 - Gorillaz.
JSexton - I was thinking of This is Radio Clash as soon as I saw the thread title.
There is also the Old 97s song “The One” which name-checks the members of the band as participants in a bank heist.
I’m not sure how any of those fit…
The partially dissolved Clash had a song called “We Are the Clash”.
Same sort of idea, Clarence Carter in “Strokin’”
Chaka Khan, “I Feel for You.”
Heeeeeeey, Bo Diddley!
Canned Heat, Fried Hockey Boogie
Damn, how’d I miss that one?
Heh, you may want to retract that in a second. 
Big Country kind of did it in their hit song “Big Country”* from their album Big Country
*ok, ok, the real title is “In a Big Country”, but it reads better my way.
“We are the Pipettes
We will drop you in our nets
If you haven’t noticed yet
We’re the prettiest girls you ever met!”
In Luckenbach, Texas, Waylon Jennings name-checks himself, then Willie Nelson name-checks himself.
In Strange by R.E.M., Michael Stipe name-checks himself.
For some reason when I was coming up with my examples I was specifically wondering if REM had done it. And I’m not particularly a fan or anything so it was a weird thought for me to have. Very happy to see this come up - you were meant to post it! 
haha I just listened to that album last week; saw them play back in 1983 with another great band, Wire Train, opening.
Like father, like son:
In Dear John, Hank Williams name-checks himself.
**
Hank Williams, Jr.** name-checks himself in Family Tradition.
On a separate but related note, Jason Ringenberg of Jason and the Scorchers name-checks himself in Broken Whiskey Glass.
In Peru, it’s very common for traditional groups/singers to name-check in basically every song
Roy Orbison name-checks himself in Mean Woman Blues.
Jerry Reed name-checks himself in When You’re Hot, You’re Hot.
Interesting! Do you know of other cultures that do this?
And all the good people down in Tennessee
Are diggin’ Barefoot Jerry and C.D.B-The South’s Gonna Do it (Again)-Charlie Daniels Band
Back around '76: “Ballroom Blitz,” by Sweet; the singer asks the other band members, by name, if they’re ready to go.