They can trace the device ID to the specific router in the house and the name and address on whoever is paying the monthly cable/FIOS bill.
What they also don't want to share is that they won't go after a single set - because the television is the device to facilitate consumerism, where the viewers are sold to advertisers, every day, all day. The carriers gave away smaller flat screen televisions for a while when customers signed up for cable or high-speed internet service.
You are right. Big screen TVs.