It is interesting how the Sesame Street one is officially considered a seperate character, but Sesame Workshop rarely acknowledges his name as Bip Bipadotta. The credits on Everybody's Record lists him by name, but I've read that the record sleeve isn't a gatefold and doesn't include images to inform people who the singers are. And I've read that the song was recorded especially for the album, premiering on the show afterwards. And I'll be surprised if there were ever any sketches where Bip refers to himself by name.
Most official sources seem to incorrectly call him Ma Nah Ma Nah or some similar name. The video pages at sesamestreet.org all list him as "Mah-Na-Mah-Na Muppet". The 40th anniversary book does refer to him as Bip Bipadotta, and mentions that he's "hear don the album Everybody's Record" (perhaps listing a source for the name, though the way it's written is a bit awkward).
It's interesting how his appearance has changed a bit over the years (in part because he was an anything muppet used sparringly). Since Muppet Wiki I've been aware that a different Anything Muppet base was used in Everybody's Song, but after watching the sketch on sesamestreet.org I never really realised how different compared to other 1980s appearances. I had seen the segment many times as a child, and I think I'd watched it online a few times in the last few years, but didn't really notice until last week.
In that appearance, he has a shaggy ponytail. And his beard is really small. And he has short sleeves (of course after this appearance he had short arms), and had some dark green furry things on his wrists as well as his neck. After noticing that I noticed that in other appearances that the cuffs of his sleeves and collar appear to be a darker shade of green, though I'm not sure if those are seperate pieces or not.