Did King Cyrus the Great's subjects have equal rights and freedom like in our constitution?
It is probably not correct to say that Cyrus the Great's kingdom afforded its subjects anywhere near to the kinds of rights and freedoms that we take for granted today. This is not to say that Cyrus was not remarkable in his day. It is just to say that we should not compare the rights his subjects had to those we have today.
Cyrus did, it would appear, give his subjects freedom of religion. This can be seen in testimony from such places as the Bible. He is also said to have been relatively enlightened in the way he ruled his empire. He maintained an empire that was comprised of many different religious and ethnic groups. To do so, he treated them equally and allowed them to participate to some degree in ruling themselves.
So, Cyrus was an enlightened ruler for his time. He treated his subjects of various ethnic and religious groups respectfully and equally. But this is not to say that he ruled over a democratic society or one where common people were equal to aristocrats.