Is it legal for my wife’s credit card company to call me, discuss her balance, and ask me to pay it?
No, it is not the proper way that a debt collection company may act in an effort to collect on a debt.  Under the Fair Debt Collection Practices Act, a debt collector may not discuss the particulars of the debt with another party.  They can call to verify the debtors address - and they can call neighbors and relatives generally only once - but they are not supposed to reveal the debt or the collection proceeding. Violations of the act can result in serious penalties and fines.  You can check on-line for the act and the parameters on collection.  And I would also contact your state attorney general's office immediately to file a complaint.  Your wife, of course, is the proper party to do so.