They didn't worship the same gods exactly. They all worshipped very similar gods. The Greek gods are a developement from very primal deities, but they are a logical development. All cultures in Europe tended to develope a female goddess of fertility and a male god of war etc.
These gods aren't the same but they are similar enough that when the Romans came along they simply assumed they were the same gods because the gods could choose how to present themselves to mankind.
In the ancient world religion was about observance to the gods and action, not about faith and dogma. There was no Roman "Bible" to say exactly how the gods were.
In a lot of ways pagans were far more tollerant than Christians, Muslims or Jews. Pagans belived that all the gods existed if you worshipped different ones that was fine. As to why Christians were persecuted, well they refused to worship the Imperial State Religion, Emperor Worship. The issue was really political, they also went around telling everyone else they were going to hell.
Bookmarks