Google-chrome – Login with Chrome to two different AWS accounts at the same time

Firefox has an extension that could do what you want (have separate accounts logged into a website at the same time, and uses separate set of cookies for each "profile").

Multifox is an extension that allows Firefox to connect to websites using different user names. Simultaneously!

For example, if you have multiple Gmail accounts, you can open them all at the same time. Each Firefox window, managed by Multifox, accesses an account without interfering each other

Each Multifox window is flagged with a number indicating the identity profile. Logins made in windows with different numbers are isolated.

Logins are preserved the same way they are in “regular” windows. Even if you close the window or quit Firefox.

The identity profile of each window is preserved when Firefox restores the session.

It can also easily switch between profiles:

For most sites, it doesn't matter: you lose the cookies containing your login session, so any future attempt to access the site will treat you as never logged in (usually same as logged out).

However, this is not necessarily the only thing that can happen on logout. You could be holding server-side resources locked (e.g. a record only one person may edit at a time) that will not be unlocked until either proper logout or the session times out. This can take many minutes, hours, or even days depending on the app. Another case is if your account can only be logged in in one place at a time.

Therefore, overall, it is better to log out if you can, but for most sites there won't be a difference. Banking sites usually don't care.