An amplifier takes a low power input signal, and a supply of power, and outputs a higher power version of the input signal, with some specified level of fidelity. The output power comes substantially or exclusively from the power supply, and is almost invariably higher power than the input signal, though less than that taken from the power supply. The output can have higher voltage, or higher current capability, usually both.
Although a transformer can increase the voltage or the current of a signal, it can never increase the power. The output power comes exclusively from, and is strictly less than, the input power. I would be very reluctant to describe what a transformer does as amplifying voltage or current, it transforms or changes these values.
There is an unusual active device called a magnetic amplifier, which was very popular before high powered silicon devices were available. It uses magnetic saturation in specially wound transformers. The input signal controls the saturation, which is then able to control the throughput of a much larger supply of power to an output. It is not a simple passive transformer.