Is it possible to apply for a marriage green card after it been working illegally and paying taxes with my real social security?

Yes, assuming you are married to a United States citizen. Your unauthorized employment will not bar you from adjustment of status. You will need to disclose this employment when completing the marriage-based adjustment of status package. The only issue you may face is if you claimed to be a United States citizen on any employment applications or the Form I-9. You can read more about adjustment of status at http://myattorneyusa.com/adjustment-of-immigration-status.