Saying the "equilibrium shifts" is a little misleading. You are correct that the equilibrium constant (Kp in this case) will be constant at the same temperature. What we mean by a "shift" is that, in response to a change in pressure (volume), the reaction ceases to act as a dynamic equilibrium and experiences net progress in one particular direction. This is because the stress upon the system (changed pressure) changes the reaction quotient Q such that Q ≠ K, and thus net shift in the relative amounts of reactants and products occurs until Q = K and the reaction is again in dynamic equilibrium.
For example, let's saw your reaction has a Kp = 1. The reaction will initially be at equilibrium where P = 1 atm for each species: Q = (1 atm HCOCl) / (1 atm HCl)(1 atm CO) = 1 = K.
If we double the volume of the reaction vessel, we see the P drops to 0.5 atm for each species. Now, Q = (0.5 atm HCOCl) / (0.5 atm HCl)(0.5 atm CO) = 2 ≠ K. Now Q > K, and the reaction must shift to reestablish equilibrium.
Solving with the quadratic we can find that after the shift the P_HCOCl = 0.382 atm and P_reactants = 0.618 atm for both HCl and CO. Now we see:
Q = (0.382 atm HCOCl) / (0.618 atm HCl)(0.618 atm CO) = 1 = K. The reaction is again at equilibrium, but now P_product ≠ P_reactants.
