It could be a label thing. Kind of like how Michael Jackson couldn't use his own name or singing voice on that episode of the Simpsons (Stark Raving Dad) because of contractual agreements with either Sony or Pepsi or both. On the note of Jackson, there IS a version of "Lisa, It's Your Birthday" that Jackson performed himself if you're interested.
As for Green Day, I think it's more like they hid their identity on purpose rather than trying to evade Reprise. What's interesting about Green Day from a label/business perspective is they have a lot of creative control. How they managed to get it like that is a different story but apparently, Billie Joe is quite a shrewd guy when it comes to that. One record executive compared him to Axl Rose saying that he's "someone who wants complete control". But of course Billie said at the time that if it's your music then why would you want a label to tell you what to do with it? Record labels really aren't your friends. The best way to think of them is like a bank that more or less doesn't charge interest.
I hope they keep doing stuff like this in the future. It's very entertaining.