1password's Windows version does run under Wine, including the browser extension. It's been a while since I did it, but I think there was some sort of browser extension validation feature that had to be disabled. Not 100% up to the security standards of their other platforms, but it's functional.
I used it (1P) and it was super, but mac only - no Linux client. Just switched over to Enpass, and its very like 1Password, only they do provide a linux client. So far its great, very happy with it.
I was really surprised that I was able to get 1Password for Windows working under Wine, even 1P Mini works with the browser extensions. It's not terribly reliable, though.