We don't even get the question to save the password or not on Chromium based browsers. I have tested Chrome 93 on Fedora 34 and Windows 10 and Edge 93 on Windows 10. Things work fine on Firefox on Fedora 34 and Windows 10.
Chromium-based Edge might have a hint here, it complains in the browser console that our password field doesn't have "autocomplete" attributes and suggests that it should have "autocomplete='current-password'":
> [DOM] Input elements should have autocomplete attributes (suggested: "current-password"): (More info: https://www.chromium.org/developers/design-documents/create-amazing-password-forms) <input class="pam_prompt" type="password" name="pamresponse" value>
Looking at the page linked in the message I get the impression that this should only be used by external password managers.. But perhaps something has changed over the years and Chrome now expects this attribute in order to present the question.
Hmm... This area seems fragile. With ThinLinc 4.14.0 I at least get "Show all passwords" when I right click the password field. After bug 7908 that entry is gone though. And adding the autocomplete fields doesn't seem to change anything.
Firefox doesn't seem to care about any of these changes, though, and always figures out the correct thing.
Tested with Chromium 105 and Firefox 102.