It's a bit more complicated. Both come from related albeit different Latin words. TL;DR:

• Fascism - etymologically "bundleism"
• Fajitas - "little strips"

Fascism

From Latin ⟨fascis⟩ /ˈfaskis/ "bundle, package, parcel". Usually of thin things, like rods. The plural ⟨fasces⟩ was specifically used to refer to a specific bundle used by magistrates.

In Late Latin times it likely got a more regular form like *fascius, inherited by Italian as ⟨fascio⟩ /'faʃʃo/ "bundle, bunch". Then you got fascists using it to refer to their own ranks, so you got "fascista" and "fascismo".

Spanish inherited Latin ⟨fascis⟩ as ⟨haz⟩ /aθ/~/as/ "bundle, beam". Note that /f/→/h/→Ø is common in Spanish.

Fajitas

Diminutive plural of ⟨faja⟩ "strip, ribbon". Ultimately from Latin ⟨fascia⟩ /'faskia/ "strip, ribbon, band".

⟨Fascia⟩ is clearly related to ⟨fascis⟩ but not quite the same word: ⟨fascia⟩ refers to a single thin object, while ⟨fascis⟩ is a bit of a collective term for the whole bundle.

The direct Spanish descendant of ⟨fascia⟩ is ⟨haza⟩ /aθa/~/asa/, that refers to a strip of land. The form ⟨faja⟩ is likely a side-borrowing from another nearby Romance language; Wiktionary says that it's from Aragonese ⟨faxa⟩ /faʃa/, it seems sensible.

The word also pops up in Italian as ⟨fascia⟩ /faʃ.ʃa/ "strip, band".