Just to clear up the question and the answer - Your Mac shipped originally with a particular version of the Mac operating system. You may install any newer version you like, if it is supported by the Mac hardware. Any full version upgrade ( 10.3 to 10.4 / 10.3 to 10.5) is a purchased upgrade, so you can purchase that software and install it legally.
You DO need to be careful what you purchase, so you don't waste your money and time. The software that came in the box with a new MacBook can't install on the old Mac mini.
Yes, 'twould violate the user license, but it will refuse to install on your old system, even if you tried.
If you want to upgrade your G4 mini to Leopard, you would purchase that software. You can buy it from Apple. You can also find it on eBay, etc, but you need to be careful that you don't purchase a grey disk set, which would leave you in the same situation. A grey set is for only the system that it shipped with originally. If you purchase through eBay, make sure it's a black disk, which will indicate the commercial install, and you should be OK.
Just to re-clarify - OS X does not have license issues similar to Windows activation, and is not serialized in any way. Someone may mention OS X Server, but that's a whole different question, with a different set of answers.