I was presuming that they would be linked via a mortgage, but you're right, I shouldn't have assumed.
If the account is in his name, he'd be liable, however the debts get run up. Even if the card with your name on spent all of the money. He'll have signed in the initial agreement to say any debt run up by any cardholders is his responsibility, too.
I'd double check that he has taken them out in his name and not yours, though. I can't see why he wouldn't have told you, otherwise. He went to the effort of ordering a card in your name, so why didn't he give it to you?
You can use Equifax to look at an online version of your credit file within a few minutes, and that'll show if there are credit card searches, or a new account, in your name. It'll also show if your financially linked. There is a free trial if you haven't already used it.
At least that way you'll know for sure whether the debt is his responsibility, or if he's applied in your name.