I came across a similar issue during the beta test for version 12. There were some old components that earlier versions of Mathematica used that were not removed during a simple installation.
So, now, I routinely remove the $UserBaseDirectory (etc. -- depends on your installation history) before any upgrade. You should move this directory (folder) to another location so that you can copy back any customizations you have installed, such as stylesheets or applications or packages.
Since I did this, there have been no issues, and macOS Mojave reports that not only is Mathematica fully 64 bit, there are no legacy apps or components associated with Mathematica.
There is a new web page (which Dale has linked to) about uninstalling Mathematica. This is not the "official" web page, and recommends the use of a different application.
The official support page:
Uninstalling Mathematica
refers to $BaseDirectory, which no longer exists for most users, but is otherwise pretty thorough. This is the method that I use. You will need to re-enter the activation key (or restore the file that contains it), and then copy back any stylesheets, etc. that you want to keep.
Once you have gotten rid of the "cruft", you can probably install new versions by simply replacing the app. I did this for the final version of Mathematica 12, and everything worked just fine.
Note that this is probably a general issue for upgrades of many apps, that there may be older bits that are no longer used, but which are still on disk. The problem is more obvious now due to the 64 bit issue. (I have had to do a similar thing to other apps that I have upgraded over multiple versions of macOS and multiple computers.)
Further, if you encounter any weirdness with Mathematica, I would recommend un-install and re-install including removing the $UserBaseDirectory. If nothing else, it will restore your installation to a "known state" for dealing with tech support.
