Thanks for all the pointers. Interestingly the upgrade went smoothly with my 2nd user account, even `xdg -open enpassauth://dropbox/` worked without any additional work.
However, this didn't work on my main account but I found a solution by simply manually invoking Enpass:
/opt/enpass/Enpass "enpassauth://dropbox?state=security_token_sinew&code=<code>"
All arguments (state and code) are shown in the browser URL when trying to set up Dropbox sync.
With this I could set up Dropbox sync successfully.