Apple - Cannot Upgrade Xcode 11 to Xcode 11.1

As per Xcode 11.1 Release Notes, it is a known issue when trying to update to Xcode 11.1. To solve the issue, use the suggested workaround -- delete the existing Xcode.app or temporarily change the Xcode.app file extension.

Known Issues

Xcode may fail to update from the Mac App Store after updating to macOS Catalina. (56061273)

Workaround: To trigger a new download you can delete the existing Xcode.app or temporarily change the file extension so it is no longer visible to the App Store.

I had the same problem. Try to drag it to the trash. You might have to purge some Xcode files in ~/Library because the App Store thought it was still installed. This worked for me.