Using the selection button tool and selecting one point from a shapefile selects all points in that shapefile. Possible to select single points from the attribute table, Select by Attributes to select a single point, and will select a single point in editing mode. This is a new quirk. 10.3.1
Trying to recreate this (colleague in another state having the issue) and the only way I could was by setting the Selection Tolerance ridiculously high (default 3 pixels).