If a region has multiple dynasties associated with it, the dynasty is used for clarity. Many of these eastern regions are so ancient that we have to consider many dynasties when thinking about history. If you want to talk about Persia, there is a string of dynasties with different characteristics going back millenia, all on the same location but often ruled by different peoples. Achaemenid, Sassanid, Safavid, etc. Knowing which "Persia" we are referring too is very important. Many of these eastern places are named after peoples as well. Iran refers to the Aryans, Turkey to Turks, and Arabia to Arabs.

We do the same thing for European countries. France refers to the Franks, but you can talk of Bourbon France or Napoleonic France; or Tudor or Stuart England just as easily as you can talk about Seljuk or Ottoman Turkey. There are also names for all these places that derive from earlier peoples, such as Gaul or Britain. Asia Minor was called Rome by the Byzantines and Muslims before the Turks changed the ethnic make up of the region. Persia refers to people from the region of Fars, etc.

We should also consider the vast geographic distribution of Arabic speaking peoples. They cover so many territories it is much more clear to talk about the Ummayad, Abbasid, or Fatimids than it is about a single region. Of course, place names are sometimes associated with these dynasties, like referring to Fatimid, Ayyubid, or Mamluk Egypt. The conditions of those dynasties were all vastly different, and they were all only within a few centuries of each other! When you consider the Ancient, Greek, and Roman, let alone the Ottoman and colonial history of Egypt, you can start to see why referring to the dynasty can be a huge factor in clarification.

I think this is a bit of a misnomer. The Sassanid Empire is also called the Sassanid Persian Empire just as frequently, while there are other empires in Europe, such as the Angevin Empire or Napoleonic Empire that were named after the family. The biggest thing to consider as to why historians might use a dynastic method of labeling comes from the need to differentiate different states/empires. So for example, one could label the Ummayad Empire as the Arab Empire, or one could simply call it the Caliphate but neither of these are particularly specific and do not do much to differentiate between what are very different polities. The Caliphate of the early Ummayads was much different than the Abbasid Caliphate, while the Arab Empire of the Ummayads was much different than the Arab Empire of Saladin. The fact that the majority of Eastern Empires centred around dynasties makes this an easy way to differentiate these different states; more often than not when those dynasties fell so too did the Empire, as the Empire itself was built around the dynasty. That said, the first point is the most important. I can't think of any 'Empires' that are named for the dynasty that aren't so named to avoid confusion with earlier or later states in the same area. Despite the Neo-Babylonian Empire all being based around the XI Dynasty (of almost certainly all related individuals or individuals who claimed relation) but we don't refer to it as the Nabopolassarid Empire; there wouldn't be any point. However, if someone said the "Chinese Empire" they could mean any one of several states spanning more than three thousand years of history.

On the other hand, if someone says the "British Empire" we are all pretty clear on what this means. By that token, saying the "French Empire" generally is going to be assumed to mean the later colonial empire, as opposed to Napoleon's Empire.