I’m happy with Mullvad Browser but there are certain sites I’d like to stay logged in to. I’m wondering if a hardened Firefox (or default settings if recommended) + Mullvad Browser Extension can give me the same privacy protection as Mullvad Browser with default settings can.
You’ll benefit from the privacy protections of the extension just fine. However, you will not have the same level of fingerprinting protection on Firefox as you won’t be blending into any crowds, unlike with Mullvad Browser. You’ll definitely stand out more if you manually harden Firefox or use something like Arkenfox.
Mullvad browser is essentially Tor browser but without Tor, so you’ll get all the benefits of the Tor browser but without being locked to using just the Tor network. It’s most effective with Mullvad VPN since most people using Mullvad browser are going to be Mullvad VPN customers.
The things that make Mullvad Browser Mullvad Browser are mostly (or fully) unrelated to the browser extension.
Mullvad Browser is downstream of Tor which is in turn downstream of Firefox ESR (extended support release). The primary value Mullvad Browser provides is strong and uniform anti-fingerprinting protection out of the box, in addition to strong privacy protections.
While technically speaking most of MB’s features are Firefox features, and you can probably achieve a mostly similar configuration with Firefox ESR, it is strongly not recommended because (if your priority is anti-fp), trying to DIY it will almost certainly end in a configuration that may be close on paper, but make you stand out in reality, because you still won’t adequately blend in with Mullvad Browser users (and blending in with one another is the primary reason to use Mullvad browser). At least this is my working assumption unless/until I see concrete evidence to the contrary.