Rather than just finding the closest point, allow the Near tool to calculate the distance between a point and it's matching pair in a separate layer based on a matching unique ID. Or, if the relationship between the tables is not one-to-one, modify the Generate Near Table tool so it can calculate distances for tables that have a one-to-many relationship.
Right now, the only way I can find to calculate the Euclidean distance between an origin table and a matching destination table is to use the XY To Line tool, which requires me to create a table with the origin and destination coordinates. It's a hassle to do the up-front work when it seems there should be a more direct way to calculate distances between point feature classes.