Detail like websites remembering you and log you in automatically is stored in cookies, so make sure you aren't clearing important cookies.

You can use these steps to make a website recognize and remember you.

• create a cookie allow exception with the proper protocol (https:// or http://) to make a website remember you

You can check that you aren't clearing important cookies.

• using "Delete cookies and site data when Firefox is closed" to clear cookies keeps cookies with an allow exception
  in 102+ version toggling this setting makes changes to the "Clear history when Firefox closes" settings and those settings prevail
• using "Clear history when Firefox closes" in Firefox 102+ honors exceptions and keeps cookies with an allow exception, previous versions removed all cookies

Make sure to keep the "Site settings".

• clearing "Site settings" clears exceptions for cookies, images, pop-up windows, and software installation and exceptions for passwords and other website specific data

• Settings -> Privacy & Security
  Cookies and Site Data: "Manage Exceptions"
• Settings -> Privacy & Security
  Firefox will: "Use custom settings for history":
  [X] "Clear history when Firefox closes" -> Settings

Thanks for your feedback and suggestions Cor-el and Dropa.

I don't believe that it's just a coincidence that the 2-factor authentication keeps asking for verification beyond the initial sign-in ONLY since the latest Firefox update.... It's logical to note that if only one variable changed (the Firefox update) and the result is different, then it's not ALL those other websites that are suddenly at fault. It's far more likely to be a change in Firefox, no?

And thanks for the steps Cor-el but I expect my browser to work as well after updates as it did before and am not prepared to fix it on behalf of Mozilla by following all your technical steps for every website that has stopped working properly. Time to re-consider Chrome.

Thanks again for your input :-)