I am using Firefox 102.0.1 and am having a similar problem. I delete all history, including cookies when I close a browser session but have listed certain sites in the "manage exceptions" link under settings>privacy and security>cookies and site data. I have been able to log into the URLs listed there without requiring dual-factor authentication. This seems to have stopped for some sites, including google.com since I received an update to Firefox. After doing some research I found that there are add-ons that can protect these cookies. I installed Quick Cookie Manager, listed as a recommended add-on, and entered two of my problem URLs in to protect them. I am still having to respond to a request for further authentication because my computer is no longer recognized by these sites.

I don't no what changed, but today the cookies are protected and I don't need to add another authentication factor.