The -id (or sometimes a small variation, like -ite) indicates lineage. The Ummayads trace descent from Ummaya, the Abbasids from Abbas, the Hashemites from Hashem and so on.

Not all Muslim historical states were named in this fashion. The Sultanate of Rum, the Mamluks and the Almohads would be counter examples to the trend.

As for why it's used, well, it's used because it's useful. Even if their territory was similar, the political discontinuity in Egypt and the Levant makes it worthwhile to speak of Fatimid Egypt, Ayyubid Egypt, and Mamluk Egypt, and so on.