If I have IDX for site X, and IDY for site Y, then there is no cross over of information there. If I have a single card IDZ for site X and Y, then they have a single piece of shared information IDZ from which to data mine. That's the unwanted linkage problem
Plus if I get IDZ stolen from site X, I've also lost it for site Y. That's the 'molehills into mountains' problem. It makes each breach more serious.
Since its common to TWO sites it's at a higher risk of being stolen, the weakest site is the one easiest to lose it... weak link in the chain problem.
I don't see the gain, I do see a lot of needless pain there. I'd prefer to have 2 factor authentification.
Single ID card for bank logins? No thanks.