The Encyclopaedia of Islam entry on him (and some of the other sources I've found) is annoyingly unclear on this, but at a minimum the short answer seems to be "no", he was not usurping "Shahanshah" to signify that he was emperor, or for that matter that he was Caliph. I'm not actually aware of later Caliphs using that title. The Ottoman Caliphs, for instance, were properly "Sultans."

The Encyclopaedia of Islam entry notes:

AL-AFDAL B. BADR AL DJAMALl, ABU 'L-KASIM SHANANSHAH [sic], Fatimid vizier, commonly known in history by his vizierial title.

It's not clear to me whether this is suggesting that "al-Afdal" was merely his vizierial title (which had precedent and continued, his son was also called "al-Afdal"), which would make Shahanshah his proper name. (Abu al-Qasim is a kunya, basically a nickname given on the birth of your first male child.)

It's perhaps worth noting that there might be some word play going on here. In Arabic "al-Afdal" means "the favorite" or "the most favored". It's not quite grammatically correct in Arabic but by placing "Afdal" in front of "Shahanshah" they might be playing off of a construction that suggests he was "Favorite of the King", i.e. his vizier.

It's also perhaps worth noting that both he and his father were of Armenian origin, which might explain the odd naming conventions.

Confusion about what his actual name was aside, I don't think I would read too much into the "Shahanshah" aspect of his name. It's barely mentioned in the sources I'm looking at.