A lot of comments here are focusing on whether Airbnb can "own" BnB. That's not quite the issue. The interesting question is whether consumers would see "BNB" as indicating commercial origin, or simply as a descriptive abbreviation for bed and breakfast accommodation.
The fact that so many people in this thread immediately interpret it as the latter is one reason I described this as an uphill battle for Airbnb.
Failing to defend, failing to adequately resource a defence of even the blindingly obvious can lead to some odd outcomes.
Personally I hope for a pro bono, well structured "piss off, this is stupid" line of reasoning from history. Ask yourself why "AirBnB" chose the 3 letter form in the first place. Thats .. prior art, a generic, or a combination of the two.
(not a lawyer. Hence my doubts this is as straightforward as I personally would like)
But may be it's the "n"?
Worth noting the "They won’t let us compromise by using a different case, an ampersand or an ‘and’" bit - so it's not just bnb which they claim to be infringing, it's also BnB and B&B.
Companies House lists thousands of businesses using those letters, and there'll be many more which aren't limited companies. Plenty of them will predate Airbnb's formation. Despite the difference in legal firepower, I suspect Airbnb might find it hard to win this one.