The famous verse is, "Render unto Caesar the things that are Caesar's, and unto God the things that are God's." Many modern writers have interpreted this passage from New Testament as a basis for secularism in modern era.
There are various, but here's one example from contemporary scholar, Bernard Lewis,
If the idea that religion and politics should be separated is relatively new, dating back a mere three hundred years, the idea that they are distinct dates back almost to the beginnings of Christianity. Christians are enjoined in their Scriptures to "render ... unto Caesar the things which are Caesar's and unto God the things which are God's." ... it has generally been interpreted as legitimizing a situation in which two institutions exist side by side, each with its own laws and chain of authority—one concerned with religion, called the Church, the other concerned with politics, called the State.
Here's also another one from Fustel de Coulanges' Ancient City. Interestingly, he contrasts Christianity with the Roman religion (relating it to Roman state worship),
Christ teaches that his kingdom is not of this world. He separates religion from government. Religion, being no longer of the earth, now interferes the least possible in terrestrial affairs. Christ adds, “Render to Caesar the things that are Caesar's, and to God the things that are God's.” It is the first time that God and the state are so clearly distinguished. ... But now Christ breaks the alliance which paganism and the empire wished to renew. He proclaims that religion is no longer the state, and that to obey Caesar is no longer the same thing as to obey God.
It is understood that secularism is a product of modern era. But, back in antiquity, was there any interpretation from Christians that see this verse similarly to modern writers? I'm particularly interested in knowing if there was any Christians who see this verse as a contrast between Roman religion and Christianity.
Let me say that in any question like this, is is always important to recognise that "state", "religion" and "separation of state and religion" are all concepts that are constructed in philosophical world views that developed later than the Biblical text. So we need to exhibit caution about how we find and apply parallel ideas.
That said, I want to also suggest that for Judaism at the time, there was an intertwining of national/ethnic/religious identity that was difficult to 'unravel'. I think this is why, in the canonical gospels, you see regular misunderstands in which Jesus is depicted as a political messiah, and he consistently rejects this interpretation of his own role.
Anyway, to turn to your question about Matthew 22:21 and its parallels, and especially interpretation in antiquity, I'm looking at the 'Ancient Christian Commentary on Scripture' series, which is a commentary that simply collects and gives you representative quotes from early Christian interpreters on a text. It seems to me, reading across the comments not only on Matthew, but Mark and Luke, that the early interpreters do not use this in developing a political theory, but rather more individually and devotionally to believers, in terms of their obligations to the Roman state and their obligations to God. This allows writers like Origen, Justin Martyr, to emphasise that Christians are to submit and obey the state, even while their higher obligation is to honour and worship God.
It's interesting that de Coulanges' quote goes where it does, because while on the one hand I agree that the Roman state and Roman religion are intertwined, post-Constantine, and especially post-Theodosius, that union is put back together in a new way. Especially in the Byzantine tradition the notion of the Emperor as heading up a new Christian world leads to very interesting overlay of religion and state.
The difficulty with interpreting "render unto Caesar what is Caesars" as a call for secularisation was that Christians never had any part in government until Constantine came to power and Christianised the empire. It certainly didn't help that the "church" such as it was was a hodgepodge of people with differing beliefs who couldn't even agree on the nature of Christ. After and during Christianisation there was an aggressive movement towards the church and state becoming one, for example Theodosius decreed Paganism illegal in favour of Christianity. For example, the Patriarch of Constantinople and for some time the Pope were appointed by the emperor and the emperor had to intervene frequently to help resolve doctrinal disputes, eg Councils of Nicaea and Chalcedon. Hell! Constantine even called himself the 13th apostle, it seems difficult to get much farther from a separation of church and state to me.
I hope that this answers your question.