I think that would be very hard to crack down on. Hotels are listed on multiple booking sites and have their own site as well. As a property owner I would never want to be stuck with only one rental channel, if one tried to force me I would simply stop using it.
Hotels/hostels are not supposed to not list on AirBnB I thought. (I stayed at a very low-end hotel booked from AirBnB in, and I was very disappointed. When I book an AirBnB I hope for a real host, conversation, see how they live and what I got was a stupid hotel... but I digress.)
They can choose to delist the properties that have a separate channel, by crawling the web, or just querying google, query, etc. to find out if the listing is just a shell. Or they can look at the messages to see if the host suggests book instead on this website. (That of course doesn't prevent direct email communication through which the same outcome transpires. It just catches the most naive showroom cases.)
There are lots of 'condotels' in tourist areas, which are essentially full-service condo buildings that explicitly sell on the idea that the owner can use it for their own holidays, then rent it on AirBNB/VRBO in all the other weeks of the year. These condotels almost always have their own booking site.
I’ve definitely stayed at B&Bs that I booked through other means and I found out by chatting with the owner sometimes used AirBnB as well. No idea what is officially allowed. But if I were renting I’d need a real good reason to grant an exclusive.