I don't really think the Ummayad Caliphate was like the Holy Roman Empire in which they were more like confederation of tribal groups and small Kingdoms or states. The Ummayad did have more influence and exercise more control over its subjects and territories, whether it was gained by peaceful submission (such as in Egypt), military conquest (like in this case Spain), or through other means. I'd rather call them more as territories with "limited autonomy" who are faithful of their Ummayad government. However like you said, it might have similiar characteristic with earlier Persian Empire, although I can't be sure since I don't know much about Persian history.

Control, as far as I know, was mainly through the spread of Islam, installation of trusted Ummayad people in high offices, and also through marriages with local rulers or people which they had subdued. About this last point, basically the Ummayad tried to gain trust and loyalty from the numerous people it encountered. Be it Berbers, Persian, or Egyptian. I read one article in the past, that even in Spain, supposedly one of the Arab military generals married a local Spanish ruler (or the other way around) who later converted, through this they managed to calm and gain trust from the many non-Muslim Spanish subjects.