What happens when primary and secondary coil of an isolating(1:1) transformer are connected together(in parallel)?
If you connect the two windings in phase, then you get an inductor. It has the same value as the inductance of either winding by itself, but it is effectively wound with copper having twice the area. As it's based on a transformer, it's a pretty lousy inductor, with a very low saturation current, and very low energy storage for the amount of iron used.

If you connect the two windings in anti-phase, then you get (for an ideal transformer) a short circuit. The inductance is zero, as there is no magnetic field for any current. No magnetic field means no change of magnetic field, means no back emf, means zero impedance. You have a short circuit, to both DC and AC.

In a real transformer, a DC signal experiences the winding resistance. An AC signal experiences the winding resistance plus the residual leakage inductance due to less than 100% coupling of the two windings.

This anti-phase connection of a transformer or coupled inductor is used in two principal places. As a transformer for detecting unbalanced currents in ground fault interrupters. And as a coupled inductor for putting inductance in the common mode of a signal or power feed pair, without filtering the differential signal present.