You can look at Multi-Account Containers.

• https://support.mozilla.org/en-US/kb/containers

This is neat, but it doesn't suit my purposes. I would rather have the problem on the site fixed rather than circumventing it.

Such special features are usually stored in cookies, so you need to make sure you allow cookies for X.

You can check for issues with Total Cookie Protection.

• https://support.mozilla.org/en-US/kb/total-cookie-protection-and-website-breakage-faq
• https://support.mozilla.org/en-US/kb/introducing-total-cookie-protection-standard-mode

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

Disabling Enhanced Tracking Protection/Total Cookie Protection did not fix the issue. "Clear history when Firefox closes" and "Delete cookies and site data when Firefox is closed" are both disabled.

Does it still happen in a new profile? An easy way to test a new profile is to install Developer Edition and see if it happens there or refresh your existing profile. Make sure you stay signed out of your sync account when testing.

The problem still happens in Developer Edition. I have tried Refresh Firefox but it does not work.

Uninstalling and reinstalling Firefox does not work.

I have tried logging into multiple accounts in a new profile. The account switcher did not work on the profile.

Does it work with the ESR version (115)?

It does not work. (115.9.0esr)

Did you check the Web Console?

• https://firefox-source-docs.mozilla.org/devtools-user/web_console/

This is what the output shows when switching.

This problem still occurs. Any help?