By selecting a single quad, you’re simultaneously selecting 4 edges and 2 perdendicular ‘flows’. One flow of edges (say, both horizontal edges of a quad) runs from top to bottom, another flow of edges (both vertical edges ) runs from left to right. That’s why you get two perpendicular rings after clicking Select > Edge Ring.
To get only one ring, pick a single edge that’s oriented perpendicularly to the ring that you want to select, and then click Select > Select Loops > Edge Rings (see pic).
