Right. They would necessarily have different orbital periods because the size of an orbit is directly proportional to the time it takes to complete one orbit: https://en.wikipedia.org/wiki/Orbital_period#/media/File:Sol...

This is why objects in geostationary orbit can only exist at a particular distance from the earth: https://en.wikipedia.org/wiki/Geostationary_orbit

Well, that's not the issue here. The issue is two planets can't exist near each other because they can't possibly have formed that way 4.5B years ago and kept existing at that point since then. Their gravitational influences would have long since caused them to collide into each other and form a single planet. There's a reason there's big gaps in the orbital distances of every planet; it's simply not possible to pack them in too tightly.