Regarding saving password in browser, the author probably means an encrypted password manager such as KeePass or 1Password, and not the current implementation which does not have a master password
Firefox has a master password that encrypts the data. I dont use it, but doesnt a Sync account also sync the passwords between instances of your browser?
That said, can you still copy the profile to a new computer and have it autofill? Physical access to data on a hard drive is often end of game, unlike cloud only hosted stuff that streams. Im guessing you cant copy a firefox profile with lastpass or 1pass or dashlane and have the new computer start autofilling without reentering the master password?
Firefox also has Firefox Lockbox now for storing passwords in the cloud.
I've noticed Safari on iOS offering to generate a password and store it when I register for new sites recently. It works pretty well (until I'm on Windows and need the password).