There are precious few similarities between the political systems of the two civilizations; in fact they were almost diametrically opposite each other.
The Persian Empire was ruled by a King from a central location; however the size of the Empire was such that he could not effectively rule alone. He employed a number of governors called Satraps to rule in his stead in various provinces. The Satraps were almost always Persian, but he did allow local officials to be chosen from local people with local customs and language. To prevent insurrections or treason, he employed a secret police system known as the "eyes and ears of the King" who provided him with information of events in the far reaches of his Empire.
The Greek poleis were each independent of each other and governed separately. Sparta was most similar to the Persian Empire as it was ruled by two Kings. The Kings were descended from separate families and did not often agree with each other. They could and were often overruled by a ruling council known as the Ephors. Athens was a direct democracy in which all adult male citizens were required to participate in the government. Some other poleis were ruled by monarchs, others by the ruling nobility. Others were ruled by politicians or generals who did not gather power by normal means. They were commonly called "tyrants" because of the method by which they gained power. In almost every instance, the only commonality between the Greek poleis was that the spoke a common language and had a great disdain for all thing foreign. In fact, they often complained that non-Greek persons sounded like the noise of sheep when they spoke: Baa-baa; hence the term "barbarian.