Why is the pressure between the interface of two liquids is same?
In general it is not the same. If the interface is not flat, the difference in pressures (Laplace pressure) can arise from non-zero surface energy of interface. Or if there is preffered curvature of the interface (that can arise from highly assymetric molecules), additional terms in pressure difference can be present.
However, if the interface is flat and simple, the equivalence of curvatures arise from the force balance. If one of the pressures is higher, it would move the interface until force balance is restored.