Cloning does what cloning means: an exact replica of the record.
When you click on a clone record button you are in the form showing values for all fields from the cloned record, but this new record does not exist yet. At this moment you can modify certain fields - for example e-mail address for the contact has to be unique - or clear context of other fields, like in your case you can clear the content of the Short Name field - and finally save it.
While it is possible to do it in the code we can not change the general code just for you. Record Browser's cloning function iterates through all fields, including custom ones (added by the user). It is impossible to exclude some fields from cloning - what criteria would we use?