Your Yahoo! ID is stored as a cookie on your computer, but only one Yahoo! cookie at a time can be stored on a computer.
If you go to a Yahoo! service and find that the Yahoo! ID displayed is not yours, you should click the "Sign Out" button and then log in with your own Yahoo! ID and password.
If you would like to have multiple users or multiple accounts on the same computer, just click "Sign Out" to log yourself out at the end of each session. In Mail, click "Sign Out," and then click "Sign Out Completely." You can then log in with a different Yahoo! ID and password, or register for another account.