I believe the closest functionality would be Capture Fit. Once you have placed a component in the assembly with relationships, select the component and run the Capture Fit command. A dialog will appear showing the relationships on that component that can be captured. You can remove any that you do not want to be remembered. You can choose whether or not to save the captured relationships in the component file so that they will be remembered any time you place that component from that point forward. If you choose not to save the captured relationships in the component file, they will be remembered until the document containing the component is closed. Once you have captured the relationships, you will only need to specify the target component / face every time the component is placed or pasted into the assembly. The only caveat is that not all relationship types are supported.
Hope this helps,
and besides all the given answers, which all of them are good and correct (I also would prefer either Capture Fit or Steering Wheel) please do not forget that we have System Libraries too.
Defining a system library not only allows You to save the relations (I don't like the SW word mates), but also to have multiple parts connected to each other and to bring features (holes, cutouts) into the parts too.
For the shown example my personal best methode anyway would be a pattern, using the hole feature or if not that simple You also can use the "Clone" function of ST10
All together seems much better and easier than those copy with mate.
But as always, this only is a personal opinion.