A TIN is issued for tax purposes only and only to select groups: certain non resident or resident aliens, their spouses, and their dependents.

It is not the same as a SSN.

This is DIRECTLY from the TX DMV website and is exactly what you need to do when you go into the DMV next time:

“Individuals who do not have a SSN, have never been issued a SSN, or are not eligible for a SSN can visit any driver license office and complete an affidavit attesting to such. Providing a false statement in connection with this affidavit will result in the cancellation of the driver license or ID card and may result in criminal charges. An affidavit is not acceptable for the issuance of a CDL.”

Do not accept no next time. Tell them you were told you need to complete a social security affidavit. That it can and has to be done then and there, in their presence. Best of luck