A portable "installation" is meant to be portable, i.e. installed on an USB-Stick and carried to use on several computers. Some people use a portable installation not to have any traces of it in the system and so don't have to uninstall but only to delete a folder after the decision to not use the software anymore.
Regarding this it does not make any sense to use a portable installation and then want to have the possibility to use a shell-extension for easier use, because such a shell extension always needs an entry in the system registry. It's just that Windows works like this.
So, if you don't want to install MP3Tag regularly with this feature or change the registry manually, just live with the consequences.
I personally can see no reason why not to install MP3Tag regularly, especially if you want this feature.
BTW: It is not necessary to have 2 screens to use drag&drop.
Windows is able to show more than 1 window on the screen.
Therefore it is called "Windows".