Xcode 7 update completes but Xcode 6.4 remains

You can download the DMG separately this time. You do not have to go through the App store, but it requires a free Developer account.

1) Log into the Mac Developer Center
2) Scroll down to the "Xcode 4.6" section.
3) On the right there is a button to download Xcode 4.
4) Open the DMG.
5) Drag the old Xcode from the Applications folder to the Trash.
6) Drag the new Xcode from the mounted volume to the Applications folder.

This time it will download a DMG file instead of a link to the Mac App Store.

Either way you will have to download reinstall any extras from within Xcode itself. See the Preferences -> Downloads tab to finish the install.

Okay, solved: I just clicked the "Purchased" tab in my App Store, and it looks like it was installing the entire time.

This was extremely confusing, because for some reason there was absolutely no indication in the Updates tab that Xcode was installing or downloading (as there usually is for updates), so I didn't even realize that it was installing. There was also no indication in the "App Search" Panel where I got the page for Xcode with the install button on it, i.e. when I clicked the button, the only thing that happened was the button changed to "installing", and there was no progress bars or statuses anywhere in sight, so I wasn't sure if it was actually installing or just hanging.

I just realized, since I DELETED the entire Xcode the very first time, when I'm installing Xcode it doesn't count as an Update, so that's why it does not appear in the Updates tab. I guess when I searched for Xcode in the App store, and clicked Install, it was installing in the Purchased tab.