This is an open issue on the Mullvad Browser GitHub.

One person in the thread who’s using Pop OS posted this:

xdg-settings set default-web-browser start-mullvad-browser.desktop

However they do report the clicked URL not being passed to the browser, so it doesn’t seem to be that good of a solution.

This reply worked for someone on Arch though.

I’m having the same issue, so I will try to fix it later and get back to this post if any of these methods work.

EDIT: I tried both methods. The Arch one did nothing, while the xdg-settings one did set Mullvad Browser as my default, but when I click a link the browser will open on the homepage anyways. So yeah, seems like we gotta wait for this to get fixed by the Mullvad devs.