Hi,
I have checked this with a Marvel Champion tester for FFG (that, specifically, was a tester for the SHIELD expansion) and he has confirmed that SHIELD applies to both the ally and the support. In fact, the official Spanish translation is like this: "Cuando se derrote esta carta: el jugador enfrentado descarta 1 Aliado o Apoyo S.H.I.E.L.D. que controle." Here, you can see that "SHIELD" is AFTER "Apoyo" (Support) so, if this trait were to be applied only to one of the two types of cards, we would have an inconsistency between the Spanish card and the English one (in Spanish, only the Support should be SHIELD; in English, only the ally).
Thanks in advance!