For mobile safari Bitwarden (and I think a number of others, but Bitwarden's the only one I can speak to) ties into Apple's password management system for autofill and password generation. Still have to use the app or webpage (either Bitwarden's official site or self-hosted vaultwarden) for more in depth management.
For mobile Firefox, on iOS it's the same as Safari. On Android you can either use the Bitwarden add-on or use it with the app and Android's built-in password management system just like on iOS.
Since you mentioned "all browsers" for chrome/chromium based browsers there is also on add-on for both mobile and desktop. For Internet Explorer and pre-chrome Edge I don't believe there's an add-on but it can still work, it'll just be more of a pain since you autofill either won't work or will be spotty. You'll probably be relying on the standalone desktop app.
On MacOS it integrates with Apple's password management, so no need for an add-on on desktop safari.
For other browsers, you'll probably have to use the desktop app and manually copy/paste just like for IE.
I also remember seeing some third-party integration for the windows terminal app and various Linux terminals, but I can't really speak to their quality or functionality since I haven't used them. But that would probably cover your needs for terminal based browsers like Lynx.
There's an add-on for the browser for both, but on Mac, the desktop app is what integrates with the system wide password manager. I don't know if desktop Firefox is integrated into that, so you may need both the add-on and desktop app to get the same systemwide functionality.
On Windows it's worth having both the browser add-on and desktop app installed as well, since the browser add-on only works in browser but the desktop app, while somewhat hit or miss whether or not it works with any specific application, is supposed to provide autofill/generation capabilities anywhere you have username/password field.