I wonder if there just isn't much profit in it for others to bother. When we had the NHL license many years ago it was already hard enough negotiating with the leagues for a decent deal, and then even harder after that to turn a profit. Hence it was inevitably dropped. I figure maybe EA is one of the few companies willing to hold on to NHL games even if they aren't very profitable just to complete their sports franchises.
The odd thing is, as far as I know, that NHL is selling more each year. The cost of producing the game has probably gone up, but I'm sure they're very economical about it. Ultimate team is popular, and that is a micro-transaction game. Maybe it helped from sinking the franchise. The season ticket thing and micro-transactions for stat boosts are also popular amongst the most hardcore players. They're definitely trying to get the average spending per customer well above the $60 price tag. The whole deal is probably not hugely profitable, if at all.
My guess is EA doesn't have exclusive rights, it's just that no one else wants to pay for licenses to the players and teams.