Hello!
I purchased the application directly from the app store but would like access to my license number because I have other devices on which I would like to install the program. How should I proceed?
If you bought via the Mac App Store, everything regarding your license is transparently handled by Apple and the Mac App Store. There is no license key or registration number in that case.
Usage is not limited to one computer and it's even possible to be used by your family via family sharing — or by an imaginary family member in case you have a second Apple ID on your other devices.
On your other devices, make sure you're logged in at the Mac App Store app with your Apple account. You're logged in, if the icon on the lower left shows you name. If it doesn't show your name, you'd need to login via ⌘0 where you can also see all of your purchased apps and can download Mp3tag via the respective icon.