Is there a way to set the language it translates to? Now it translates to the language Android is set to but I'd like to choose another.
Discussion
Add the language you want to change it to to the Android's Language and input. Then the option will be available
Seems my Android only can have one language at a time.
Strange, which android version are you using?
Android 11 on a Xiaomi.
Super strange, it says here that it should be possible: https://www.techbone.net/xiaomi/user-manual/add-keyboard-language
Yes I already have English and Japanese there.

Ok, so you do have two languages, but they don't show up when you click in Auto on a translated message?
No just the currently set system language. I temporary changed the system language to English which let me change the target translation language to English as a workaround.
Then changed the system language back to Japanese.