Neat question! Earth has a unique equator because of its shape. You could cut it in an infinite number of ways with the cut going through the center of the earth. Any way that you cut it, there would be a line around the sphere where the cut intersects its surface; this line is an equator. If earth was a sphere, all of these lines would be the same length, and you could call any one of them the equator. However… earth is not a perfect sphere; it is an ellipsoid (a squashed sphere; see the attached figure). It sort of bulges out in the middle, so it is widest at the equator. If you cut the earth with a bunch of different planes that go through the center, the equator would be the longest line on the surface.