I have a theory that it should have a very different "personality" (probably more like writing style) depending on language because it's an entirely different set of training data

In English chatGPT is rather academic and has a recognisable style of writing, if you've used it a bit you can usually get hints something was written by it just by reading it.

Does it speak in a similar tone, with similar mannerisms in other languages? (where possible, obviously some things don't translate)

I don't know a second language well enough to have natural conversation so I'm unable to test this myself, and may have worded things awkwardly from a lack of understanding