Keychain Won't Remember Router Administrative Password
The 'forgetfulness' is likely to be intentional.

Safari won't offer to remember any passwords etc if the page tells it not to.
I don't know the technical details of how this is accomplished, but banking pages etc won't do it either.
The 'remember' box on the page is not for Safari to remember, it's for the remote device.

There used to be a freeware extension called 'autocomplete' that would disable this [lack of] functionality in Safari, but it hasn't worked so well in recent years.
I just discovered the author now has a paid version, called StopTheMadness, still fully supported, which might do the same thing. I haven't been able to fully test it yet - just got it 5 minutes ago.