Is it correct to say that H2SO4 is an acid in this reaction?
Do you get the same reaction with sodium sulfate?

I do not think so, at least under "normal" conditions.

When the sulfuric acid acts as an oxidizing agent giving off $\ce{SO_2}$, the excess oxygen comes off as oxide ions which must be somehow put into more stable form.  The protons from the sulphuric acid do that by turning them to water.

Getting the reaction to go requires the sulfuric acid to act as both an acid and an oxidizing agent.