Creeque Alley sort of qualifies. While it does not mention the group (The Mamas and Papas) by name, it mentions each of the members by name and gives backgrounds and drug induced states at the time of the recording.
The Who did “Who Are You,” but that’s not really a band reference. However, in their song, “The Punk Meets the Godfather,” Daltrey refers to himself as “the punk with the stutter” and then sings “My-My-My-My G-g-g-generation.”
The Fifth Dimension has the line “Sing along with the Fifth Dimension” in the song “Let the Sunshine In.”
Nirvana, “Paper Cuts”…the chorus is “Nirvana” repeated several times.
Pearl Jam’s “Dirty Frank” doesn’t mention the band by name but mentions guitarist Mike McCready getting eaten.
I think Green Day has an early song called Green Day.
I know there’s more, older and newer ones too but that’s all I can think of right now…
David Allen Coe in “You Never Even Call Me By My Name!” (“But the only time, I know, they’ll hear David Allen Coe, is when Jesus has his final…JUDGEMENT DAY!”)
And as any C/W fan knows, this was in fact the PERFECT C&W song!
I think Limp Bizkit mentions their band name in just about every song they’ve ever recorded. At least, it’s a running joke in our house whenever they come on the radio.
Roomie: “What band is this?”
Fred Durst: “IT’S LIMP BIZKIT!” (ad nauseum)
I saw Jon Spencer Blues Explosion open for the Beastie Boys a couple of years ago, and the entire chorus of one song performed was “JON SPENCER BLUES EXPLOSION!” screamed over and over again. It really made me dislike the band.