Well, that is already tax fraud correct? Someone already has to check for that. The savings come from all the additional checks you would no longer need to do to see if people are really disabled, what their marital situation is, where they live, etc.
In most of European countries, people has an ID card and SS number, and there is a centralized database controlled by the state. A single person cannot have 2 different IDs. At least, it is not something trivial to do.