Transformer and Thevenin Theorem
That's the difference between an ideal transformer and a real transformer. The transformer equation is a very crude model of a real transformer.

Whenever you combine an ideal model (transformer) with a "real" model (the voltage source) you may get results that don't make sense, particularly when you start asking about the "physical meaning".

If not current flows through the primary then no current flows through the resistor, so the voltage drop across the resistor is zero, so the voltage across the primary is equal to Vg, so the voltage across the secondary is Vg. Vg = Vs.