Yahoo Email

Yahoo is doing that based on the fact that you have previously authenticated an AT&T e-mail address.   You can try your browser's incognito mode for your Yahoo address.  Or use a different browser (on which you have have to clear cookies first).

Why should I have to clear cookies every time I use a browser?  That's absurd to require people to do that!!!  There needs to be a different solution.

Using incognito mode is not clearing cookies.  Using a different browser is not clearing cookies.   As for why it's necessary, that's an issue with Yahoo's authentication mechanisms... take it up with Yahoo.

I am trying to help out my 77 year old father who has one account for business and one account for personal emails.  I tried keeping one browser logged in to one account and another browser logged in to the other account, but they continue to revert back to the AT&T one randomly.  I even tried bookmarking the site and creating shortcuts on his desktop, but every time it reverts back to AT&T.  I haven't tried incognito mode yet, but I'm afraid that may be more than he can understand.  There has to be a better way.  Yahoo says to contact AT&T for support, so "take it up with Yahoo" is not an option.  

Should you get an official AT&T response, it will be "talk to Yahoo."  The web interface belongs to Yahoo, they code, maintain and the responsibility for supporting it is theirs [though, yes, when AT&T white labels it for their use, they can assume that responsibility but don't].  The Yahoo web software has to decide how to authenticate you, i.e. whether they should get your e-mail address and decide "oh, that's ours; we need to authenticate you ourselves" or decide to 'remember' that you've used an AT&T address in the past and jump over the "how should I authenticate you" straight to sending you to AT&T to authenticate you.

So, again, as a Yahoo mail address owner, Yahoo isn't giving you a chance to log in as a Yahoo address owner, deciding to instead send you to AT&T for authentication.  This is a failure of Yahoo.

Another way to deal with this might be to set up a mail client program for one or the other e-mail addresses.  Windows Mail will work with Yahoo (both regular @yahoo addresses and AT&T addresses), so you could use it for either or both.  If you don't like Windows Mail, there's Outlook.  I even think Outlook 365 could be used.  If you're using a mail program (like all of the above) that doesn't have Oauth2 authentication support, then you'll need to get an App Password (Yahoo) or Secure Mail Key (AT&T) to use in lieu of the password in client setup.