“Is Romans 13 about obeying civil government or church government?”
In the context, it is true that Paul just finished speaking about “he that ruleth” in the local church (Rom. 12:8). But the “rulers” in Chapter 13 bear a “sword” (Rom. 13:4) that cannot refer to “the sword of the Spirit” (Eph. 6:17) borne by church rulers, for he that bears it is called “a revenger to execute wrath upon him that doeth evil” (Rom. 13:4). That doesn’t fit the role of rulers in a grace church, but it is an apt description of civil rulers. In the more immediate context, Paul has just finished quoting God as saying, “Vengeance is Mine; I will repay” (12:19). So when Paul then goes on to call the bearer of the sword “a revenger,” he is explaining that God takes vengeance on evil doers in the dispensation of grace by the sword of civil government.
In return for their civil service, we are told to pay these rulers “tribute,” something paid to kings (Matt. 17:24,25; 22:17), not rulers in grace churches. The Jews paid tribute to Israel’s religious leaders (Num. 31:37-41) because they were a theocracy, a government ruled by God, and so paying tribute was just giving God His due. But the word “due” (Rom. 13:7) speaks of what is owed as a debt, and so “tribute” is a word that cannot be used of giving in the local church. Under grace, our giving is not done “of necessity” (2 Cor. 9:7). source