How do password managers know when I've logged in successfully?

They aren't actually aware of a successful login in most cases. They are aware that a form with a password field was submitted, and the response was a 200OK. This may still be a page displaying an error message.

As for extracting user IDs, I'm pretty sure you mean profile pages or something similar. That will have to be done on a site by site basis as sites will have their own APIs and route structures.

As someone already answered this question, I will agree with him.

They aren't actually aware of a successful login in most cases. They are aware that a form with a password field was submitted, and the response was a 200OK. This may still be a page displaying an error message.

Since browsers watch for the request having a password field in it, and also the response status code of the request, But still you can fool the browsers easily. To get to know about the logged in userid you definitely need backend support / api. It depends on the authentication frameworks used in the back-end. But you can get the form fields easily, but extracting / finding userid from the form fields is a quiet difficult task, In most cases form will be having only two fields there you can manage to get the userid. But in some cases like banking sites they will send few dummy fields and fool such tools. In some cases userid is different from email, So its difficult task.

They only detect if the form was submitted, and it a code 200 (OK) was returned. They don't necessarily know if you were logged in, but this method works on most websites. They might also detect if a new page was loaded afterwards, since a failed login doesn't usually redirect the user. I have, however, had a prompt to save an incorrect password before.