The debtor claims that the debt is just one of a series of debts charged to their account by someone who stole their wallet several months ago. You are not the first person who has called about this, the debtor gets several calls a day about this stolen identification. That's a very plausible story, and in some cases it may even be the truth.
This is a case of what we call the diffusion claim. By getting other people involved, the debtor tries to spread the blame to others and if they are clever, they will even convince the collector to call the other parties to settle the dispute! Sometimes its a mother or brother, or cousin, or son. Sometimes they have fled the state, or claim there is a legal injunction to getting in touch with the other party.
The solution is to call the bluff. Ask for a copy of the police report and don't let them snow you. If it is a legitimate claim, they will have called the police and have documentation to back up their claim. In any case besides theft or fraud, the diffusion should be ignored. If they blame it on their mother, tell them to get the money back from their mother after they pay you. Let the case proceed till it gets to the law suit phase and let the debtor prove their case in court. If they are really not responsible, they will be able to provide solid documentation and be glad to do it.