Why is the change in enthalpy computation for a change of state done like this?
You're confused because the equation is incorrect.  It should read $$\Delta H=\Delta U+\Delta (PV)$$If you insist on writing it the way you did, then P should be $\frac{P_1+P_2}{2}$ and V should be $\frac{V_1+V_2}{2}$.  That will give the same result as the first equation I wrote.