Can an object's mechanical energy be equal to its gravitational potential energy?

Energy is not a physical unit. It is a philosophical concept and an accounting technique used to analyze mechanical and chemical exchanges. In nuclear physics it is defined as a wavelength of light, in biology it is a synonym for metabolism, and in public utilities they say energy when they mean connectivity. Anybody who uses the word in any other context does not know what he is talking about.

Mechanical energy is the sum of potential energy and kinetic energy. If an object is immobile but elevated, all of its mechanical energy is stored in gravitational potential energy. Therefore, yes; an object's mechanical energy can be equal to its gravitational potential energy.