Photo by NASA.
Of all the planets and moons in the solar system, Pluto and Charon are the two which resemble each other the most closely. They are almost the same size, and they are very close together. They are so close together that they orbit the center of mass between them, and the center of mass goes around the sun. (In general, each planet with its moons orbits the center of mass between them, but in every other case the center of mass is the center of the planet, so the moon goes around the planet. Not so in the case of Pluto and Charon.) Pluto and Charon are so close that they may even share an atmosphere. Methane molecules may be drawn off Pluto and recondense on Charon.
The other planet and moon in the solar system which could be a binary planet are the Earth and the Moon. As in the case of the Earth and the Moon, the origin of such a system is hard to explain. The explanation which seems best is that some large object struck Pluto (and the Earth) and tore Charon (and the Moon) from the crust of the planet. In the case of the Moon, the Moon seems to be made of material very much like the Earth's crust and not like the Earth's more metallic interior. Likewise Charon may be made of more icy material which resembles Pluto's surface.