I’ve exactly the same problem on both Chrome and Firefox. I have not tested with others.
When I submit a username / password from any website the passwords app does not attempt to remember it.
I have created a video to make the problem more clear:
The browser extension does not automatically save new passwords, it stores them internally and sends you a notification when it detected a new one.
The “+19” means that in the fourth tab of the extension popup, you will find 19 passwords that the extension detected. You can take a look at them and save them if they’re correct.
There is also a setting in the extension settings to save a password immediately when you click the notification about a new password.