I create circles in my project using the out of the box circle tool, but then I densify those circles immediately after they are created. But know I need to know if that densified shape that i created is a circle? Is there a way to get all the points of that densified polygon and see if they are all equidistant apart. This would tell me that the polygon is a densified circle.

A circle in any implementation by esri that I have seen is an 'n-gon' a 36 (or other multiple) sided shape on the perimeter of a circle. The points on that circle are connected by straight lines, so densifying them simply adds point(s) on the straight line and not at points of the same distance from the centre that the original points were. In that context, densifying the circle does little (nothing) to improve the appearance of circularity.

If you are concerned about the points being equidistant from the center radiating at some specified angle (ie 0.5 or 1 degree) increment, then you can generate them yourself. Implementations exist on geonet (ie python)