Reading between the lines: Nothing belongs to Caesar, everything belongs to God.
Right you are.
Coins with Caesar’s picture on them belong to Caesar. Scholars (Scott Hahn?) have pointed out that there were “graven image”-free coins circulating in Judea, but they traded at a lower value than the ones with the picture of Tiberius.
By holding the coins with Caesar’s image, the Pharisees demonstrated that Mammon was more important to them than the Commandment.