Hide My Email works in two ways.
The one with associated passwords is tied into Sign in with Apple. That path has no password, other than your current access into your Apple Account. (No, you’re not giving your Apple Account passwords to other websites. See thr Sign in with Apple article details.)
The other way Hide My Email works is as a mail-forwarding service, and Apple doesn’t have passwords specific to the email address. The website gets the Hide My Email address, and whatever password selected there, and that password then gets stored in Keychain associated with the website. This is much closer to how Keychain works ~everywhere, where the Keychain entry gets the username (an email address) and a user- or Keychain-provided password.
Which is a lot of words to get ‘round to not having a password showing for the website in Keychain probably means Sign in with Apple was used here.