Two different charges, q1 and q2, are placed at two different locations, one charge at each location. The locations have the same electric potential V. Do the charges have the same electric potential energy?

No

Explanation:

Electric potential is the work done to bring a unit of charge (1 C) from infinity to a point inside an electric field.

Electric potential energy of a charge q is the energy required to keep it in an electric potential V. Electric potential energy is given by,

U = qV

Hence even if the two charges are on an equipotential surface (surface where the potential is the same at all points), the potenial energy will be different if the magnitude or nature of the charges are different.