Why is it that Naruto get a cloak from the Nine Tails while the other tailed beasts dron't give one to their Jinchuriki?
This doesn't make sense unless the other tailed beast aren't as powerful as the Nine Tails or is it that we never see the other tailed beasts show themselves in this form with their Jinchūriki. I just want to know if the other tailed beasts can grant such a cloak to their Jinchuriki or not.
Many times throughout the series Naruto did manifest a cloak in the same way so the other hosts:
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.
The cloak is exclusive to Naruto and Minato because they are
father and son.
The cloak we see them use is not Kyubi's chakra but their own chakra. Only the edge of the cloaks is the Kyubi's cloak. Naruto's cloak is surrounded by a small whitish edge and Minato's by a blackish edge. Those parts are the yin and yang halves of Kurama.
When Hashirama notes that Naruto and Minato gave a massive amount of chakra to everyone, he said,
Their chakra is almost as much as mine.
then he continued,
No, there is Kyubi's chakra here as well.
This means that most of it was Naruto and Minato's chakra.
Remember that Naruto had to forcefully take Kurama's chakra in order to get chakra mode. But after that, my assumption is that after gaining cooperation with Kurama, Kurama gave him an upgrade to chakra mode to make him even stronger. So I think all jinchuriki could have a chakra mode if they did the same thing as Naruto. It has nothing to do with the yin and yang sides
Only the form shown in the question is unique to Naruto. Killer B has been shown to have cloaks similar to Naruto's 1 - tailed cloak (as in the picture I've uploaded). The first time Naruto got the Nine Tails Cloak shown in question was when Naruto was training with Killer B. Naruto fought with the Nine Tails and forcefully extracted the chakra from it. So, this is a form that appears when Naruto taps into that reserve of chakra he stole from the Nine Tails. None of the other Jinchuriki have this form because they either have complete co-ordination with their tailed beasts (like Killer B) or don't have any such reserve of chakra. 
According to Naruto wiki these are different types of tailed beast transformations.
Initial Jinchuriki Form : The jinchuriki's body is shrouded with tailed-beast chakra.
Jinchuriki Form version 1 : In this case the chakra starts taking the form of a beast and tails start manifesting.
Jinchuriki form version 2 : In this form, depending on the number of tails, a user' body is shrouded in boiling blood. Bones may also start manifesting if the number of tails increase.
Partial Transformation : The jinchuriki can transform any part of their body into the tailed-beast equivalent of the same.
Full Transformation : The jinchuriki takes the form of the tailed beast.
Nine tails chakra/cloak mode : This form is said to be unique to nine-tails jinchuriki.
My speculation is that this could be the result of Naruto having only the yin-part of Kurama because of which he gets the form. The only other person to don this form is Minato Namikaze, Naruto's father and unsurprisingly he too had half-tailed beast inside him.
So either it could be a unique form of nine tails jinchuriki or it could be due to having only half of Kurama inside that led to this form being activated.
This is late, but it's because of his Eight Trigrams Seal. Only Naruto and Minato had the Eight Trigrams, because it was a special seal created and used by the Uzumaki clan primarily for such a purpose, Kushina use her Adamantine Chains, so she never needed it. Also, only Minato and Naruto could manifest their forms as pure chakra.
Given to Naruto at birth, the Eight Trigrams seal slowly leaks Tailed-Beast chakra to the Jinchuuriki, the longer two chakra flows are in contact, the more in sync they become. Naruto and Kurama are considered perfect sync by the start of the Fourth War, the only thing that held them back was their personal feelings and cooperation. Other Jinchuurikis were either in control (i.e. Bee) or taken over by their Tailed-Beast will (i.e. Gaara). Only Naruto and Kurama had everything set up to be a perfect Jinchuuriki/Tailed Beast relationship.
The answer is simple. Bad writing. They just wanted Naruto and Minato to have a unique look so they gave them a unique look even though this made less than no sense and directly contradicted the established world building. We might as well ask why Madara as the 10 tails Jinchuuriki was vulnerable to taijutsu when jinchuuriki gaining a cloak which makes one immune to all taijutsu is one of the main advantages of being a jinchuuriki. Yes, Guy should not have been able to harm Madara even slightly because the 10 tails should have given Madara a cloak that rendered him invulnerable to taijutsu.
Also according to Naruto (the original series), the entire 9 tails was sealed inside Naruto just as has happened to every other 9 tailed Jinchuuriki. The retcon in Shippuden about the 9 tails being split in half was directly contradicted by every other arc. For one thing Kushina had the entire 9 tails sealed in her and she was 100% fine. Every leaf village Jinchuuriki before Kushina also had the entire 9 tails sealed inside them and they were completely fine. There is no evidence to support the assertion that Naruto could not or did not have the entire 9 tails sealed inside him. Bee had the entire 8 tails sealed inside him and he was 100% fine.
In this case absence of evidence that the 9 tails had been split in half before the second to last arc is evidence that the 9 tails hadn't been split in half and this was a massive retcon which destroyed the internal consistency of the series.
As far as i know....It is because the kyuubi cant manifest itself in the form that the other jinchuriki's can. If u remember... all the other jinchuriki's can convert themselves into the bijou form and revert back at will. But in naruto's case, the kyuubi isn't able to do dat conversion i.e. get into bijou and get back to human again. I think I remember der was one episode in which the kyuubi says why, not sure.
So, I'm thinking to make this up, the best the kyuubi can do is make a cloak around naruto. Cloak is just the manifestation of the kyuubi's chakra. For all that we know, naruto could have even had a cap! But to make the semblance to minato, they formed the cloak which is very similar to minato's cloak.
