Ok, I think I understand the situation now. Your phone has the Samsung Internet browser (which is Samsung's own browser app, separate from Chrome). At some point, you created a shortcut icon on your homescreen (what you're referring to as the desktop) for the AT&T homepage (which can be done by opening the browser app, going to the AT&T site, then tapping the browser menu button, then Add to Home Screen -- this essentially creates a bookmark on your homescreen). If you tap that shortcut, you're still opening the Samsung Internet browser app -- it's just that it's automatically opening the AT&T homepage.
If you tap the Samsung Internet browser icon itself, it will open the browser app either to the last screen you were on, or possibly to the homepage that you set for the browser (which is usually found in the browser's settings menu). If you find a homepage option in the browser settings, you could set that to AT&T as well.
Keep in mind that if you routinely access your AT&T homepage by tapping the shortcut on the homescreen, it might open a new browser tab each time. If you have too many browser tabs open, then at some point, it'll tell you that you have to close some. To do that, tap the tab management button (usually next to the URL entry bar), which will show you all of your open tabs in miniaturized form. You can swipe them away or tap the associated X at the corner of each tab to close them.