use GenerateNearTable using the same FC for both in_features and near_features and the 'ALL' option on closest.
That give you a (potentialy very big) table of distances between every point and every other.
You can also get the bearing angle too, if you want.
PointDistance may work for you too.
You may want to make selections on the 5K_ID field before running it, to get (many) smaller tables to work with.
you need an advanced license to use GenerateNearTable or PointDistance
AddXY_management will give you locations for each point.
You could then do some trig on the X,Y location differences between pairs of matched points
Here is a workaround idea:
1. Make sure the two sets of points are in the same projection/coordinate system; then use the Merge tool to combine them into one input.
2. Use the Points To Line tool with the 5k_ID as the Line Field to obtain lines between each pair of proposed and real points. The line output carries the 5k_ID field and the Shape_Length values would be the distances you need. For shapefile output, you would need to add a field and then calculate the shape length, using the Add Field and Calculate Field tools.
3. Run the Join Field tool to join the line output with any of the two point inputs via the common 5K_ID to transfer the Shape_Length field to it.
Does it work for you?
Have you had a look at Hawths analysis tools - Distance between Points tool:
If Anthony's or Dan's suggestion hasn't solved your issue - please let me know which version of ArcGIS you are using.