It would be like owning property / or land, and some one drives over top of your land everyday without your permission so that it cuts tracks as a rough road into your land so the land is violated but not taken over, and the police (the State) empowers those who drive over your land (your Church) even after you said not to do that.
As such the Churches said "no" to the same-sex marriages (their land) but the State law forced that the Churches could not stop that violation.
So the Churches are not taken over as the Churches are only violated.
The separation of Church from State has been violated because the State has taken control over the religious Institution of marriage.
While only the government can recognize the union of two dogs, and the government is the only one who can empower the marriage of two dogs.