I found a solution that might be helpful in these situations.
1 >>>First Use buffer tool to create a small polygon on every point of your layer.
CAREFUL! set the Dissolve type on all. or you will have multiple features in the same spot and will make a lot of trouble later!
Choose a small radius unless you want to join a small cluster of data. for example, your points are not exactly in the same spot and may differ a bit.I used 0.5 meter radius. even 0.1 meter is enough if the points and exactly the same
2 >>> Use Feature to Polygon Tool to split all the polygons and eliminate all multi-part features.
3 >>> open Spatial Join and put your points as Traget and the buffer as join feature.
4 >>> Now your points in same location have Identical join FID Field. So you can dessolve your points based on this field. open Dissolve (Data Management), your points as input, check Join FID as dissolve field. then on statistical field choose Object ID and select count as Statistic type value.
5 >>> the result hase your every point with number of those have identical locations in Count ObjectID field.
I used it to analyse the intersections and how many roads lead to an intersections.