The question is whether you want to host the server backend yourself or not and whether you want to manage the passwords independently of a particular browser or not.
If you use the integrated password managers of the browsers, they synchronize the passwords via their own servers. So Edge synchronizes with your Microsoft account, Chrome with the Gogele account etc…
Then there are browser independent password managers, like Bitwarden, Keeper, 1Password etc. that provide their own server backend as a service, and corresponding apps and browser extensions. This has the advantage that you are not tied to a specific browser, or to a browser at all.
For some of these services, such as Bitwarden, the server backend can also be self-hosted, or in the case of Passwords, it even has to be self-hosted with the Nextcloud Server App.