This does not appear to be true.

The word fascism comes from the Latin fasces, which denotes a bundle of wooden rods that typically included a protruding axe blade. In ancient Rome, lictors (attendants to magistrates) would hold the fasces as a symbol of the penal power of their magistrate. The first European fascist, Benito Mussolini, adopted this symbol both to recall the greatness of the Roman Empire and to reinforce his authority as the eventual dictator of Italy. Fascist regimes like his required their citizens to be as unified as the tightly bound fasces.

Britannica

While fajita is the diminutive of faja, meaning strip, tracing back to the Latin fascia, band.