If you generate a fingerprint of the audio on the phone, you would not leak too much information about the environment. Shazam, for example, computes the spectrum, picks the strongest peaks, and uses the relative positions of small groups of peaks in time and frequency as features to search for in known recordings. Those features are quite sparse and you can not reconstruct the audio from them. You could however identify more or less every audio signal in the environment you have a copy of to compare against.

You could do better by sending out the fingerprint you are looking for and compare it against the past couple of seconds or minutes on the phone so that you could only report a match if one occurred. This would avoid leaking what music you are listening to or what you are watching on TV unless it is what they are looking for. If you report a match with GPS coordinates, the server could throw away everything but the position so that the position is not easily linked to a user.

This still reveals all the living rooms in which someone watched the match and used the app, so it's not perfect. If you have a map of all relevant businesses, you could just count matches in the proximity for each or you could only keep matches from locations from which a certain minimum number of matches were reported which should also get rid of most living rooms. This is still not perfect, it, for example, potentially leaks how popular different places are but from a privacy perspective of app users it seems acceptable to me, at least given you trust them to do it right.

The real issue, at least in my opinion, is that they turn the app users against the business owners of the places they like to watch matches at. I am not against them trying to track the ones down that are not paying, they have a legitimate interest in that. But the way they are trying to do it seems wrong to me. There are probably some app users that would welcome if everyone had to pay but I guess most don't really care whether their favorite sports bar pays or not and even more would not want to cause trouble for the business owner even if they think they should pay. In consequence this is a feature that many if not most app users would not want to use even if there were no privacy issues. They still put it in hoping that nobody would take notice and that they could get away with it, at least for some time.