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I would imagine so. "Gotten" is the preferred form in American English although "got" is permissible. "I have got" sounds as weird to most of us as "I have gotten" sounds to Canadians, Australians, and the British.
When I first learned English, the book I used would never say, i.e. "I have a book". Instead, it was using "I've got a book". So, even though I learned I could say "I have", the form "get-got-got" was the one I got accustomed with.
When I first saw the form "gotten" I was surprised.

Anyway, about the on/by accident thing:

I had never heard or read anywhere "on accident". I've always seen "by accident".

"On accident" sounds to me like "during an accident", kinda like "An accident was happening, and during this accident he broke the vase". Still, I've never heard it before.

But hey, what do I know anyway??

See ya.
AnnaBtG.

What we've got here is failure to communicate...