The government already knows if someone has stolen your identity, and you are never told.
Wages are reported quarterly, and if you file for unemployment, Social Security, any type of government benefit, you have to report your employers, where you have worked, and wages are matched under your Social Security number. We can already see what is there, it is up to you to verify the information.
Often, a Social Security number will pull up wages for another name, employer that seems "unusual", and the wages are just removed from the Social Security number.
Surprising, isn't it? But that money floats the system, because that employer is paying taxes, Social Security, Unemployment insurance, on that number, for a person who can never access that benefit.