There is no in-built way of doing this AFAIK, however you could use MapBasic to develop a custom drawing tool that would provide this functionality.
It should be noted that this would not be a foolproof method of picking up the actual snapped-to node in the case of having multiple selectable background layers. Unfortunately there is no functionality to return the snapped-to layer or object, you would have to search the locality (within a small tolerance) of the click location for any underlying objects. Where multiple objects are returned you would have to find a way to identifying which one you wanted to use for the attribute information.
If you do want to do this, a good start would be to look up
SearchInfo in the MapBasic User Guide.