Opera and Chrome are just two other browsers. I also use Netscape Navigator (another one, which was before Firefox from the same Mozilla people) but there are dozens and lots I have never touched, for Windows/ Mac/ linux.
On Firefox, when you tick the 'remember it' box, it will be saved internally. If you look at Tools / Options and check Security (I think - I'm not using it this minute so could be wrong) you'll see a button marked 'saved passwords')
Click the button and a new window appears showing a list of websites and username(s) [in case you have more than one user for a particular website]. You have to click another button and confirm you want to see the passwords to get it to display three columns.
You can sort the information (eg by username) by clicking on the column heading. So you could check if you have the same username on too many sites...
All too many people are conned into 'registering' for some site and use the same combination of mail address and password or username and password. One example was where a BBC reported got a message on FaceBook saying there was a funny video of him...
The link needed him to register and by doing so, he gave away his mail address and password, and that info meant his mail address could be accessed. Once into his mail, it would be possible to find out which bank he uses, and by extracting lists of contacts, start attacks on other people to see if they might fall for the same trick.
When he viewed the video, it was of a pirate or something (sorry, don't have the link handy) and clued him up to his having been victim to an attempt/achieved account details 'theft' / intrusion.
I will try to find the link - I might have it bookmarked on another PC as an example for teens not to trust messages about having a funny video clip of them... all too ego- / vanity- / driven that they'd go looking...