It is absolutely the definition.

You're linking a subjective, loosely interpreted editorial of what racist fascim could present as, given 22 minutes to slog along another conclusion-driven narrative (which I watched, again), while I am providing a synopsis of the actual definition:

"a political philosophy, movement, or regime (such as that of the Fascisti) that exalts nation and often race above the individual and that stands for a centralized autocratic government headed by a dictatorial leader, severe economic and social regimentation, and forcible suppression of opposition"

Fascism is a governmental body taking control of society, disallowing individual thought and behavior.

As I said.

The punisher is uninterested in running an autocratic government that controls other people via oppression.

That makes him, definitively, not a fascist.