Sounds like they're misrepresenting what the NYC regs actually say. I would bet they are aware what they said is not completely accurate.
A driver affiliated with one could do rides for the other, so long as the dispatcher makes the passenger aware, according to TLC spokesperson Alan Fromberg.
The way Lyft is making passengers aware is by putting the vehicle's base ID on its picture on the app, per TLC requirements.
It goes to show you, if they could they'd like to force all drivers into an exclusive.
The only reason they don't is that pretty much throws the independent contractor relationship out the window and makes drivers employees. If they could use a regulation to force the issue, that would be a double jackpot...driver exclusives as independent contractors.