Andy ,try reinstalling the latest release of Firefox ,and after this even this happens go to tools and then logins ,there you can find your credentials which are saved. If that's empty try logging to any website for example gmail , it will ask for remember the password or not , give it a yes. And as you mentioned you have a Xperia device , if this problem still persists we will see to the problem and this will be further escalated from our side and you should also try reaching sony for this. Have happy browsing with Firefox.

Regards: Gaurav