Hi twic,
welcome to the Apple (user-to-user) Community!
Re: How can I make the wallpaper name appear in macOS Ventura? : Name in NEWER Systems
Wondering just where you are expecting to view the wallpaper's name.
Eg: Am using the latest Ventura Mac OS ... 13.6.6 ... on MacBook Pro:
If I go to Apple Menu > System Settings > Wallpaper (scroll down left sidebar):
Can view the name of any wallpaper image by clicking the image, and seeing the name appear in the name-box above all the images, while selected image appears on the screen of Mac image. Also: hovering the cursor above an image will display that image's name as well

If this is not working for you on your Mac Mini, you may want to send feedback to Apple:
(They will likely not reply, yet your feedback will reach those who can make changes.)
If others experience the same issue, we may see an update that addresses it.
Feedback - Mac mini - Apple
All the best :-)