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.
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.)