I have also wondered about the possibility of media emitting some kind of hidden identifier. Can normal speakers emit noise outside of human hearing that could contain a unique id? Would not even need to be a lot of data that could be broadcast.

Yes, it's possible. Some sports broadcasts do this. The ones I'm familiar with use a high pitched tone on the edge of human hearing but still audible to most people.

Streaming services watermark the video as opposed to the audio as I understand it.

But that's for tracing pirated content back to the originating user account. It's not particularly useful (also entirely unnecessary and overly complicated) if your goal is to ID a piece of content as opposed to an account.

Live broadcasts make it hard for Shazam type apps to ID something from a fingerprint, so they need something else to help.

That's not necessary. If a service has access to the raw audio of a show, it can fingerprint each second of that audio in a way that can be matched in a tiny amount of computation even for a recording in a noisy environment.

That can't recognize what you say. But, it can ID where you are in a specific show out of zillions of hours of shows.

Notably, the caught the TV also recording and transmitting recorded audio and transcripts from the microphone.

Honestly, it doesn’t even need to be outside the range of human hearing.

A quick search for “audio steganography” should answer your question (and be a fun rabbit hole if you find that sort of topic interesting)