While there are some professions that do not pay FICA taxes, it seems that your employer has confirmed that there should be FICA taxes withheld and that you are not exempt.
If you employer made a mistake on your W2, it is your employers responsibility to file a correction. They would need to correct it was far back as the error occurred. And yes, they would be allowed to recoup your share of the FICA depending on your state.
The wages not being reported for FICA does impact your Social Security earnings record.
I found some additional resources regarding FICA deductions listed below:
Social Security Administration W2c
IRS Correcting Employer Payroll Forms
Employee Deductions
