The term 'democracy' is simply one of three terms used in Hearts of Iron 4 to put each nation into an idealogical faction. (Democracy, Fascism and Comintern). It's not meant to describe exactly what the government type is.
For example:
- The United Kingdom was a constitutional monarchy whose government was consistently run either by the Conservative Party, or the Labour Party, both comparatively moderate, so they come under 'Democratic'.
- The Kingdom of Italy was a constitutional monarchy whose government was run by a fascist regime under Mussolini, hence it comes under 'Fascism'.
Persia, of course, was also a constitutional monarchy. The question is not really whether Persia would support Germany or not, or whether it had a King. It's mostly about the idealogical attitude of its ruling government and people at the time.