Could they trademark (it wouldn't be copyright) a term that they themselves copied from a work of fiction they don't own the copyright to, given that the context (i.e. what it does) isn't different to that in the work of fiction?

Yes - and yes, trademark, not copyright, pre-coffee conflation. Consider the Nexus line of phones, lifted directly from PKD's Do Androids Dream. Copyright is about area of applicability - I bet babelfish is trademarked for "translation services and products".

This is like as if I’d trademark "Quidditch" for a sports league where a sport is played with brooms, flying around, etc.

That shouldn’t be possible, and hopefully any court would laugh at whoever claims to have a trademark on Babelfish.

Oddly, while I can find three trademarks, Yahoo isn't one of the holders.

https://trademarks.justia.com/785/15/babelfish-78515910.html

https://trademarks.justia.com/772/49/babelfish-77249656.html

https://trademarks.justia.com/851/07/babel-fish-85107190.htm...