Does storing photos and music on iCloud free up storage on my Mac?

Not directly, no. iCloud is not an off device storage.

It may reduce the amount of storage taken up by them in certain cases, but will not free it up entirely.

For music, if using the iCloud Music Library, you can remove downloads from the Mac to free up some space.

But it still highly recommended to keep a backup of your music library locally on an external drive. If anything ever happens to the Apple Music subscription, anything not locally backed up can be and usually is lost.

For Photos, again, if using iCloud Photos, you can turn on Optimize Storage in the Photos App settings and it will reduce the quality and file size of photos locally, but will still take up some space.

click here ➜ Optimize storage in Photos on Mac - Apple Support

Just to add a bit to Phil0124's comments:

iCloud is a synchronization service--it's purpose is to make some of your stuff available to other devices connected to your account, so you can see the same stuff on your iPhone or iPad, for instance. Also, iCloud lets you access your stuff from any computer anywhere by using a browser. As Phil0124 said, iCloud doesn't serve as an external storage service or as a backup service.

iCloud can relieve space on a crowded device by using Optimize Storage. But that's really for things like iPhones that don't have much storage space. A very big downside to using Optimize Storage is that you can't back up files that aren't local. And, again, iCloud itself is not a backup service. Often we use Optimize for our iPhones but not on our Macs, so we can use the Mac to backup our files.

Since you are using a Mac mini, a desktop computer, then for that computer the best option is to use an external drive for your pictures and music, and then to have another external drive that backs up your system.