Is it possible that Kakashi and Obito are brothers? Neither of their mothers are mentioned, and it would explain how Kakashi awakened the Kamui after his eyes were taken.
No, they are not brothers. Obito's sharingan has the ability to use Kamui. Since both Obito and Kakashi share that sharingan, both were able to unlock Kamui.
Obito gives Kakashi his left eye, which has the ability to use long range Kamui. Obito gets to use his right eye, which can only use short range Kamui. That's why Kakashi was able to unlock Kamui. (Specifically the long range variant).
Also, Kakashi is from the Hatake clan, while Obito is from the Uchiha clan.
Since Kakashi is from Hatake clan and Obito is from Uchiha clan so I don't think they can be brothers.
Also in wiki it states that :
Kakashi teaches the importance of teamwork, a lesson he, like the Sharingan, received from his childhood friend, Obito Uchiha.
So its pretty much clear that Obito was Kakashi childhood friend not more than that.
No, they are not brothers. Obito is from the Uchiha clan, while Kakashi is from the Hatake clan.
Furthermore, them being brothers would not explain why Kakashi was able to activate his Sharingan or Mangekyou.
Sharingans work even after having been transplanted, and that is why Kakashi was able to use his — however, it is worth noting that it puts much more strain on him than it would on a "native" Sharingan user. As with any other Mangekyou users, Kakashi's and Obito's Mangekyou were awakened once they experienced great loss of someone close to them (Rin, in their case) — it has nothing to do with them being related by blood.