How do I transfer Final Cut Pro license to a new computer without losing access on the old one?

Hello Apple Community

In 2022 I purchased Final Cut Pro, which worked fine so far, also after the release of Final Cut Pro 12. Now I bought another Computer and wanted to use the same Final Cut Pro on the new computer, but there`s a message showing up, that no valid license was found and therefore I should redownload it in the app store. That`s what I did, but: after opening the newly downloaded Final Cut Pro Software, I get a another message, that I don`t seem to have a license for Final Cut Pro.

The funny thing about it: on my old computer, final cut pro still works perfectly fine without any download of new software. So: Does anybody have an idea how to fix that?

Thanks for the help!

Steve

[Re-Titled by Moderator]

Original Title: Final Cut Pro License