Identify highlights do not line up with feature

When the yellow highlight (visual association) of a feature does not line up with the feature drawn on the map, it is likely a datum shift that is introducing the misalignment. A datum shift occurs when ArcGIS Server must project a feature from one spatial reference to another.

This will happen when a dynamic service is in a local spatial reference (projection) and the map is not in the same projection - a lot of sites will be using Web Mercator as the projection since that's what the base maps use.
To check the transformations, use the Data Frame Properties panel in ArcMap for the map that doesn't line up.  Make sure that there is a transformation present for transforming from whatever the map uses to whatever the site uses - in many cases this will be a NAD83 to WGS84 transformation.
Some maps may require a transformation to be set; if the spatial reference of the map is custom and needs a transformation, you may need to explicitly add one here so that the highlight graphics will align with Web Mercator (or whatever the map uses).