Most likely not thermal expansion tank or water supply piping . What type of drainage pipes do you have . ? Cast iron or pvc ? get a 5 gallon bucket of water and dump it down the drains, if you hear noise the you have your answer. It’s the drainage making the noise …

If no expansion tank and you have a prv . a thermal expansion tank is not required by Code . As a prv can be used as a thermal expansion device .

Anti scald is required nowadays , but i have not heard of any brand making tapping sounds like you explained .
usually a mucked up Cast Iron system and hearing hiccups in the pipe from bad Venting . if pvc it could still be a venting issue but more less likely unless a bird died inside your vent stack or some form of object is obstructed in the stack . Anyway good luck , Let us know when you get it solved .

I see, there are many types, this is the one I thought you meant and can be used instead of a thermal exp tank in south Florida building code .not sure if that one has a discharge port

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.love it when a house goes into a revolt all at the same time