When you select your Mac Mini Studio from the list, you should see it twice, once for "Mirror or extend to" and once for "Link keyboard and mouse to". Make sure you select the second option. Here's an example of what you should see (it should be the same options for a Mac Mini):

If you don't see the option to "Link keyboard and mouse to" your other Mac, you'll have to go to System Settings > Displays > Advanced and enable "Allow your pointer and keyboard to move between any nearby Мас or iPad". I've tried to connect to my iPad with this disabled and I was only able to "Mirror or extend to" it.

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To be sure if it's working, you can check the current state of the display options in the menu bar (just add the module in System Settings > Control Centre). You should see this icon if it's working: 