Do you mean saving logins to the Password Manager (about:logins) ? Can you use "Create New Login" ?

• https://support.mozilla.org/en-US/kb/add-login-firefox-password-manager

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

No, I cannot seem to create a new login. I get the error "An error occurred while trying to save this password".

Try to create the same login, the password 1234567 and see if it works. see screenshot Can you share the URL you're using in your screenshot? Another test, try downloading a different version of Firefox (Dev, Beta, or Nightly) and see if you have the same issues with passwords. https://www.mozilla.org/en-US/firefox/all/#product-desktop-release

Using 1234567 as the password did not solve it - same problem. I changed my Firefox password (as it was existing since 2017) and then also downloaded the MacOS version and replaced the existing version. While replacing, it asked if I wanted to do that since I was overwriting a newer version. When I look at the "About" after replacing, it still shows the same version number but everything seems fixed now. Thank you for your help.

In the screenshot of where you tried to create a login, I see a notification bar asking to provide the Primary Password. Not having unlocked the logins via the PP is the likely cause of the error you got.

I agree, but every time I clicked on that, it would take me to my account, like I was already logged in. That's why I changed my password.