In short, password managers (including the ones built into browsers) are designed to prevent this from happening, so it shouldn't happen. If it does, then something else has gone wrong.
To expand on this, though, password managers have to find a balance between ease of use and security, and certain design choice can affect this balance.
One of the main and most important jobs of a password manager is only to pre-fill or automatically log in when it can be sure it is the same website as you were on when you saved that password. Because, if it does it on a different website, it may just inadvertently leak your password to some other site, and therefore to someone else that should not be given your password.
So to address your question, if this fake site is completely unrelated to any genuine site you have a saved password for, including a different domain name, there is no way it can get any of your saved passwords.
The difficulty is, the boundary for what defines a website can vary. In some cases, a website can have multiple domain names. Some password managers can know about this and let your login work across a network of domain names regardless of which one you initially saved it on. This is convenient, except if the password manager accidentally gets one wrong, and that one falls into the wrong hands. In other cases, websites with the same domain may be owned and controlled by totally different people, which is the case with shared web hosting where clients don't get their own domain name - sometimes they may have a subdomain with a shared domain, and sometimes even a subdirectory under the same host. Password managers can try and be smart about this by maintaining a record of domains which can be used for separate sites on different subdomains or directories. Or, they can take an overall more conservative approach and only match a site if it has the exact same hostname (including subdomain) and path to the login screen. Or, take an approach somewhere in between where if there is any discrepancy the user is prompted to confirm whether it's the same site.
Then there is the issue of site security itself. A password manager cannot know if a site's been hacked and taken over by hostile parties. It can know if a site uses https making some kinds of attacks (man-in-the-middle) more difficult though.
What all this boils down to is that there is some amount of art to the algorithm that a password manager uses to determine if the site you're visiting is authorised to be given a password you've saved before. You can help protect yourself to some degree:
- When logging in using a saved password, stop to consider whether you're on a site where you may have separate logins to separate areas of the site such as a hosting site and you're not on your own account there.
- Disable automatic login if your password manager has it (ie, where the password manager also submits the form for you instead of only pre-filling it).
- Don't necessarily just settle for the password manager provided by your browser. There are third party password managers which can have additional security features.