What is the transfer function of a transformer?
Transformer equivalent circuit: -




  We give arbitrary input voltage x(t), what is the output y(t)?


You are not considering that the input is both voltage and current and that the output is the same with a bunch of various subtleties along the way such as: -


Leakage inductances 
DC resistance (copper losses)
Magnetization inductance
Iron losses (eddy current and saturation)
Secondary load impedance


And this is just a regular low frequency power transformer.

Also would you be confused in the same way with a DC blocking capacitor and resistor forming a high pass filter?