This isn't really a browser extension. It's more of a shortcut to the desktop app.
Of all the password extensions I've used (LastPass, 1Password, and Bitwarden, mostly), Dropbox Passwords seems like more of a prank than an actual product.
For one, it feels like an alpha product more than a v1. Incredibly buggy, slow, and lacking in features. No web version of the app, either, which makes exactly zero sense given Dropbox's track record of desktop+web for pretty much all of their products so far.