"Money's all broke and food's going hungry,
If it wasn't for disappointment, I wouldn't have any appointment."
It's lines like this that make me think TMBG at their best could easily meet the criteria for a comedy band, as soul-crushing as their songs can be. That they have a wider range doesn't disqualify them in my book.
Paul and Storm is my personal favorite as they sing mainly about geek and nerd culture. They occassionally team up with nerd royalty like Wil Wheaton, Felicia Day, Jonathan Coulton, and Adam Savage. I first discovered them at PAX 2011 during the celebrity D&D game and I've been a big fan ever since.
"Write Like the Wind". A song making fun of George R.R. Martin's slow writing speed when it comes to his Game of Thrones books.
Garfunkel and Oates are a hilarious duo who sing about pretty much anything, but with a special focus on sex and the world through a woman's lens, and all of it is absolutely gold. They are probably objectively the best of the best when it comes to comedy song writing. Both of these women are also actors and appear in many comedy shows and such. I have to warn you that most of their stuff is very NSFW when it comes to lyrical content.
The Axis of Awesome is a hilarious trio from Australia. They're not well known in the US, which is a shame. Their content is pretty diverse, but lots of the material deals with making fun of popular music.
"4 Chords". This is their most popular song. It's basically a humorous and shocking look at the poverty of pop music.
"The Language of Love". This one is my personal favorite. It's a hilarious play on language stereotypes with an ending that makes me lose it everytime.
Well, not a band, but Tim Minchin is my personal favourite comedy musician. I've even had the pleasure of seeing the man perform live and he's amazing.
My favourite songs of his (All NSFW with strong language and themes):