If I purchased Final Cut Pro on my Mac mini, do I have to buy it again to use it on my Mac Studio?

I finally upgraded from my mid 2010 Mac mini to an M2 Mac Studio. Instead of going through the process of carrying over everything from the Mac mini, I have started the Mac Studio completely fresh, and I'm moving data and files from the Mac mini over to the Mac Studio and will be sorting and deleting files as appropriate.

However, I did about two dozen video projects in Final Cut Pro on the Mac mini, with all the files saved on external hard drives. I'd like to avoid having to pay $400 for the software again if I can.

If anyone has done this sort of "upgrade" I'd love to hear about it, thank you!