Web site, the same browser can only log on to one user reason (cookie cannot cross browser)

My web site, like Http://ip/testsite/default.aspx,

When I'm on my machine, open with chrome, login with account user1,
Then when I open a new tab, and then open the Web site, then the display is still User1 landing.
I want to use another account User2 login in Chrome, then can only exit User1, and then use User2 landing.
In the same browser, only one account can log in at a time.

And at the same time, in different browsers, such as one is Chrome, one is Firefox, respectively, with User1 and User2 landing, this is feasible.

This is true because cookies do not cross browsers, and we save sessionid with cookies, and the server determines whether SessionID exists.

Chrome can be created with ctrl+shift+n and will not share cookies.

Web site, the same browser can only log on to one user reason (cookie cannot cross browser)