Catholic Europe was never a politicaly unified entity with the pope as it's head. It was comprised of several kingdoms, principalities etc. that all arose indipendently from Rome, so the pope never had much controll over them all. The only place he directly ruled were the Papal States in central Italy.

The caliphs on the other hand came into beeing as heirs to the political power the Prophet Muhammad wielded over the early Muslim community. Under their leadership the islamic state conquered the territories that are marked in your third map. Later on some parts of the muslim world broke away from the political control of the center. That's why the Abbasid empire is shown to already be a little smaller. By the late 9th century most of the provinces had slipped away and the Abbasid caliphs had become mere figureheads. The leaders of the various islamic successor states still looked to the caliph in Bagdad as their nominal head but not as their actual sovereign.

The Fatimids rose up at the beginning of the 10th century as contenders for the leadership of the islamic world. Your map shows all the areas they could conquer by military might (and a bit more; the map isn't all that accurate). But there were also muslims living beyond those territories that favoured the Fatimids over the Abbasid caliphs.

So the single units you are talking of are only the empires which the caliphs acctually ruled over (and which could roughly be compared to the pope's Papal States). But those are not necessarily congruent with the entirety of the muslim world that viewed the caliph as it's religious head (and which would be the equivalent to all of catholic Europe).

If you want to know more about early islamic society and history those books might interrest you: H. Kennedy, The Prophet and the Age of the Caliphates: The Islamic Near East from the 6th to the 11th Century (2004); J. Berkey, The Formation of Islam: Religion and Society in the Near East, 600-1800 (2011)