Each person only has one "vital statistic" representing the circumstances of birth. Any document from the US, relating to Cruz, will pertain only to citizenship.
7 FAM 1441.1 (State Department Manual)
f. The Form FS-240 is not a birth certificate, such as is issued by a government- authorized bureau or office of vital statistics, because consular officers are not authorized to assume a foreign local or state vital statistics function. The Form FS-240 is a consular declaration of the fact of acquisition of U.S. citizenship at birth based upon: ...
Has he produced the form
FS-240? Did his mother apply for it on his behalf?