They do not! The reasons for this are simple: minerals are semi-stable configurations of elements formed in certain pressure-temperature-chemical conditions. A planet in the possession of active plate tectonics will also be in the possession of more extreme pressures and temperatures (not to mention introduces chemicals to such conditions that would not otherwise be present, like water and carbonates). Also, should be mentioned, Earth being such a large rocky planet (with such a large, active core) helps generate extreme PT conditions.
Also, life creates incredibly improbable redox conditions wherever it goes. An atmosphere flooded with oxygen? Crazy! Protons being ejected into the soil? Bizarre!
Suffice it to say that mineral diversity increased as life developed. The idea that life and minerals evolved together is known as "mineral evolution"... something of a misnomer, perhaps, but it gets the point across. Earth has lots of wild minerals because it has lots of wild life. New minerals are being discovered due to human activity as well; mining operations concentrates weird elements together & exposes them to warmth and humidity.
So, the answer is no--as far as we know (we have barely explored our own rocky planets & moons, much less other solar systems'), Earth's mineral diversity is unique.