For reference: Bluetooth settings on Mac - Apple Support
Your concern is not entirely clear to me but in general if a Bluetooth device is automatically connecting to your Mac, the most likely reason is that you or that Mac's Administrator approved that connection at some time in the past. Clicking the circled x will remove it from that list of previously approved devices, after which you would need to click the Connect button for it to connect automatically again. Refer to Connect a Bluetooth device with your Mac - Apple Support, and scroll down to Disconnect a Bluetooth device.
"If you don’t want a device to reconnect automatically, Control-click its name, then click Forget. You have to connect it again if you want to use it later."