Probably your feature class tolerance and resolution are too low with a shapefile to ensure that points near a boundary do not duplicate. A File geodatabase allows for double precision and I believe there may be tools to convert to high precision. In a sense it is correct to duplicate an incident on a boundary. Also, if your boundaries have overlaps, even very small overlaps, they can cause a problem.
Sloppy topology on your polygons could cause several problems. Intersecting the polygons with themselves would show you if your topology is sloppy. Making sure your topology is clean is crucial if you want very precise results.
If there really is polygon overlap or points fall on the boundary, then the other program is also skewing your results by making a random choice on which boundary to assign the count to without consulting you. I prefer the ARCGIS behavior so that I could process a secondary routine to find points that need to have their count cut in half to split the incident between to two or more areas manually (or shift its position so that it is clearly assigned to a single polygon).