How to stop Macbook from connecting to Mac Mini

I'm not sure which feature you're talking about. You could look over this list and see if there are any that appear to match the specific behavior you are seeing.

Continuity features and requirements on Apple devices - Apple Support

It wouldn't be Sidecar between two Macs.

If you see one of the Macs offering to let you use the other Mac as a display, that would suggest that you are looking at AirPlay, and that the Mac that is showing up as a possible AirPlay destination is set up as an AirPlay Receiver for AirPlay to Mac.

Set up your Mac to be an AirPlay Receiver - Apple Support

Use your Mac as a display - Apple Support

It doesn't mean that the Macs are mixing personal and work data, or that you even need to make the AirPlay connection (which will just let one Mac take over the other's screen.).

maetheunicorn Said:

"How to stop Macbook from connecting to Mac Mini: I have a Mac Mini for personal use and a Macbook for work. Both are connected to a splitter, so I can use the same monitor whether I’m on the Mac Mini or Macbook. However, because of how close both computers are, the Macbook will want to connect with the Mac Mini, which I’m wanting to avoid to keep work separate from personal stuff. I’m not sure what this is called, but how do I disable it?"

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Turning off Mirror the Screen to an External Display:

If looking to turn off mirroring to an external display, perform the following will mirror the Home Screen, and it will allow you to use it as an extended display.

• Without Touchbar: Hold Down [command + F1]
• With Touchbar: Hold Down: [command + dull screen]