The pilot initially requests vectors back to Dallas Love, but a few seconds later in the recording he formally declares an emergency and requests immediate vectors back to Fort Worth (for the extra runway length). I think the pilot only realized the issue because there is no ILS approach to 13L at DFW. It feels like this could have been a lot worse than it was. Not an insignificant miss by ATC. If they had given him 13R he might have executed an approach on 13R at Love Field.
Nothing personal, the protocols are designed so that both sides make sure comms are clear and confirmed.