I have a single application that allows for multiple user Logins from one workstation. A single user could have two accounts for this one application that would point the user to two different databases.
The problem I am facing is that when I log in with one user (A) then open a second tab and login as another user (B) the Membership and profile for both logins point to user (B).
I’m not sure how to separate the two logins.
The problem isn't on the server end, it's the browser.
You have no control over this behavior. The only option I know of to deal with it is to use one browser for one session and a different browser completely for the other. (One in Firefox, one in Chrome for example)
In my experience even if you have two instances of the same browser open (Two instances of Firefox, for example), the session is shared, so logging in as person a in one instance, then as person b in the second instance will cause the first instance to also be logged in as person B the next time a link is clicked or an action is performed.
Edit
On second thought, if this is an internal app, and you're careful to avoid session tampering, you COULD try using cookieless mode. I've never tried it because of the limitations and particularly the security concerns - it's not practical in a public-facing website, but in theory, it might work, since the identifier is in the URL, not in a cookie or browser session.