Is it normal for a bike to skid even after both brakes are applied?
To answer the question shortly: yes, it is normal.

When braking, the inertia of the rider (and the bike) causes the load on the rear tire to decrease. It leads to lower friction against the surface, and possibly skidding. In extreme cases, when braking with the front brake, the rear tire is lifed off the ground completely.

Especially in fast mountain biking, the rider must lean back or "hang" as far back as feasible when braking, to maintain better traction and prevent possible falling over the handlebar.