A rainbow appears circular because it is defined by a constant angle from the antisolar point. A fisheye lens changes how angle maps onto the sensor, but the mapping is usually circularly symmetric around the image center.

Because of that symmetry, any circle centered close to the middle of the image remains a circle after the fisheye projection. Its radius may change, but it won’t be stretched into an ellipse. What looks like “distortion” in a fisheye photo is mainly that straight lines away from the center bend; circles centered on the optical axis are preserved as circles under a circularly symmetric transform.

So the background can look strongly distorted while the rainbow still looks round. If the rainbow were far off-center, or if the lens introduced asymmetric distortion, then its shape could depart from a perfect circle.