Do diamond and obsidian have anything in common?

Yes, both are formed in the earth, both are considered 'rocks' and both have translucent properties.

Obsidian is glass that results from volcanic eruptions; diamonds are erupted to the earth's surface from volcanic pipes. Pipes are shaped like volcanoes, both with bases at the earth surface. The tip of a volcanic pipe is far below the earth surface: volcano tips soar into the earth's atmosphere.

Both form crystals, but with different structures.

Obsidian is formed from felsic lava "...which is high in silica, aluminium, potassium, sodium, and calcium..." and diamonds are formed from carbon.