Transformers do not operate the same as relays.
A mechanical relay does require a current to generate magnetic flux to move the armature. This would be true even for an ideal relay (using superconducting wire for example)
An ideal transformer, when driven by an AC voltage, draws no current if the output has no load. It would appear as an infinite inductor. There is current only when the output load draws current.
In practice, of course, the transformer inductance is finite and there is a small magnetizing current, but that does not make it a current device, at least not by usual definitions of that designation (an ideal current device has zero input impedance and infinite output impedance). The magnetizing current also has nothing to do with the wire resistance.
Whether something is called a voltage device or current device depends upon how close it is to either ideal. If it has a relatively low input impedance, such as a bipolar transistor base input, it's usually called a current controlled device. If it has a relatively high input impedance, such as a FET or standard mechanical relay, it's usually called a voltage controlled device. (Thus most relay coils are rated by the voltage they require for operation: 5V, 12V, 28V, etc.)