Wow! I've never seen those break nor even split. If anything, if you could torque it enough to do that, the clamping pressure would crack or shatter your vanity-toptub. It has to be a defect as the threads should've stripped or the nut should've cracked open before the pipe snapped.
The only thing I can think of is that the faucet handle was used as a grab-bar once...short & sweet enough that the other materials couldn't react & stayed static. If known not to be the case, then I'd have to side with getting rid of the old setup entirely.
It's to the store for a new faucet either way. You may be able to repair or replace it, as the handle appears to screw-in to the spout. The benefit would be that you MAY find a super cheap new faucet of plastic that matches right up, instead of buying a whole new double or triple the price faucet. Buy a few different ones or make a few trips & return whatever doesn't work & maybe none will. But, you can't say you didn't try.