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LG smart TVs caught logging audio with screen off and snooping on local devices
(www.notebookcheck.net)
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not necessarily, MAC is a layer 2 protocol, and your router (NAT or no NAT) operates on layer 3. Routers are supposed to overwrite the MAC address with their own when they forward a packet.
It's always possible for the unit to put its own identfying info in the payload. But you didn't do anything silly like register the TV, did you?
The dns entry uses a unique subdomain with that info hashed into it