How to prevent child from using Find My to interact with parent's device? Both me and my child are members of one Family, with Family Sharing turned on. We use different Apple IDs. When my child opens "Find My" app, my devices are listed there as well.

How do I prevent my devices from showing up on that list?

The only option I have found is that I can turn off Location Sharing. But my child can still for example use "Play Sound" to remotely play a sound on my device.

I know that I can turn off "Find My" for any of my particular devices completely, but that would also prevent me from finding this device using my iCloud account, so I don't want to do that.

Alternatively, perhaps there's a way to block "Find My" on child's device completely?
I would implement parental controls on their device. You set a passcode and then you can control which apps are hidden. 


https://support.apple.com/en-us/HT201304


Start with hiding that one app and then see if you want to impose other restrictions. You might ensure iCloud backups are set and lock their iCloud to your family plan as well for extra security in case you lose the passcode or the device becomes locked.  Some people RE prone to try to erase or get around things, so getting a good backup can save you time and grief. 

Also, you can decline to share your location with accounts one by one and let then use the app for other people to share if that’s compatible with your end goal.