It appears as though this problem is persisting through a long series of Arc Pro versions, starting at least with 2.3.3 in 2019. I am now experiencing this problem with an imagery service in Arc Pro 2.9.0. Might there be a problem saving these credentials other than with just the Arc Pro software? Some possibilities:

• whether signed into a portal (either Arc Online or Arc Portal)?
• whether saving credential to Windows Credential Manager or connection file. I just tested this and then found the site, username, and dotted password in the Windows Credential Manager, so at least this does appear to be saved and working as advertised.
• whether your chosen server connection will allow this kind of login.

I also noticed that there does not seem to be an option to not save credentials if you want to be super secret about it; there is only "save to WCM" or "save to connection".