Thanks @AaronJensen I'll be adding those instructions in the extension itself to try and make it more clear. This seems to be an issue with some Firefox settings clearing the extension's storage, but I wasn't aware of it in Chromium-based browsers.
The "manually" idea is to copy the session token from the Kagi.com settings and paste it in the extension settings. In order to get the "Save" button to work, disable the extension in incognito, save the token, and then enable it again.
Let me know if that works or if you run into the exact same issue. If so, maybe Brave will have a better error description, so if you can share any error you can see while inspecting the extension, it'll be helpful.