When you parent bones to bones, you have options of inheriting rotation, scale and location. But, I don't see those options when I do things like parent an object to an empty or an empty to a vertex. Where are these options?
Just to amplify @dr. 's answer, the 'Child Of' constraint gives Object parenting behaviour, with options for each of the basic transforms, in each of the axes.
The 'Copy XXX' constraints give more options: how to combine the target's transforms with the owner's existing ones, and in which space the copied transforms are evaluated.