Why does Naruto get a cloak from the Nine Tails while other Jinchuriki don't?
Many times throughout the series Naruto did manifest a cloak in the same way so the other hosts:


When he fought Sasuke at the Valley of the End, 1-Tail Cloak
When he fought Orichimaru after the time-skip, 3 Tail Cloak
When he was fighting Pain after Hinata got attacked, 4-9 Tail Cloak


What is different for him, is that he is a Sage, and to my knowledge, none of the other hosts are Sages. After he learned Sage mode AND reflected on his inner evil, he gained that form; see the fight with Pain, he was using Sage mode then.

Based on all of this, I think that this is something 'unique' to him (except that his father can do it too...) because of him combining Sage training and Kyuubi's power.