AFAIK, Safari has nothing like that Firefox feature. I'm inclined to agree that the site is at fault.
More info: I have an account with our local library. The login ID is my account number but the box to enter it doesn't say "login" or "user ID". Instead, it says "library account login." Safari will remember the password but not the login. Firefox, on the other hand, has no problem with either.
Jon
macOS 11.7.10, iMac Retina 5K 27-inch, late 2014, 3.5 GHz Intel Core i5, 1 TB fusion drive, 16 GB RAM, Epson SureColor P600, Photoshop CC, Lightroom CC, MS Office 365
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