Can someone explain how this is not checkmate?

Yes

If your partners opponent captures a piece ( for example a Bishop) then your opponent can block the check.

This is apparently by design in case the partner captures something to be able to block the mate

There's a forum section that's dedicated to chess variants. There you can ask questions about chess variants. For example when you have a question about a Bughouse game, which is a chess variant. You can find it here or look for it your self. The section is easily recognizable, because it's called 'chess variants'.

I recommend posting your questions about chess variants in the chess variants section, especially when you can't be bothered to mention it's about a chess variant. That way you can prevent other people from needing to open your image to enlarge as the only way to discover it's a question about a chess variant. It's how your question about a chess variant will be visible to players that share your interest in chess variants like Bughouse.