When I enter this (https://accounts.google.com/ServiceLogin?service=mail&passive=true&Emailemail@example.com&continue=https://firstname.lastname@example.org/) in browser and no other Gmail account is currently logged in, it will ask for the password for email@example.com, but if I've already logged in using another email it just open that account's inbox and does not ask me to enter a password for firstname.lastname@example.org.
That happens because
https://accounts.google.com/ServiceLogin is the address for logging in to the Gmail initially. To add accounts after there's one already signed in you ned to use
In your case it'll be: