Hey @mayanksatnalika! Sorry it’s taken me a while to respond, I’ve been quite busy the past couple of days.
Just as a side note, the browser and Gaia are almost completely separate, with the Browser only creating the tokens used to access Gaia in the first place, and Gaia not touching the browser at all.
The browser is a client-side application (hence why it can be run via a client-side website, a node.js install, or through a browser extension) that takes your mnemonic and then creates your master keychain, each identity’s keychain, and each app’s keychain when authenticating (each keychain being a public/private key pair, and the apps keychain being generated by the method I described here).
You can use your account on multiple devices/browsers, it’s just there is no “login” – there is only “recovery” via the phrase, because there is no backend to store your phrase for you! Therefore the password is just a pass-key that encrypts your mnemonic pass-phrase (NOT your identity private key, or app private key, etc due to how often they are used within the app), and nothing more.
Let me know if you have any more questions! (I learned most of this when making the extension… it was a fun process!)