The best way to know what is possible. Is to search the Web Store. Every idea you have someone already thought about it :)
And if no, you can merge two ideas into your own creative idea.
Search Google for "Multi-Session chrome web store". You will see what is possible.
According to your questions. Cookies saved per browser instance, no per tab. So you can't log in at the same time in the browser using two accounts.
BUT - Google support native multi-login, so you can do it for Google websites only because they support it.
Alternatively, you can ask the user to enable the extension in icognito window, and choose another account there, or run Gmail in the remote browser and open remote desktop, or create new user data dir, or new Chrome profile.