Does a name patch on a ESD smock also have to be ESD compliant?
Depends on what level of ESD compliance. In industries like areospace or medical, the answer would be yes, it would have to be compliant and made of ESD free material or be made conductive. For some industries, no because the cost of ESD protection is not worth the cost of the manufactured electronics.
Even though the tag would probably not be dangerous, it static electric fields could build up charge on the surface to thousands of volts depending on the material.
You could find out by testing one with an electric field meter with the appropriate impedance to see if it really is a problem and violates the ESD requirements at your locale.