OK, so having gotten past my WTF reaction and seen different reactions, there's a few things that might add up to Kakashi/Obito's Susanoo.
1. manipulating spirits isn't new
Several characters in this war arc have claimed or shown the ability to manipulate their soul. Dan and Madara both were able to perform ninjutsu when they were separated from their Edo Tensei bodies and Tobirama claimed to have the ability to manipulate this soul.
2. the Sharingan is awakened by special chakra
According to Tobirama, an Uchiha awakens their Sharingan when their brain produces some kind of special chakra that affects the optic nerves. Obito, having returned from the spiritual realm, entered Kakashi's body and his chakra awakened the Rasengan.
3. Kakashi/Obito's Susanoo is likely temporary
As Obito said, the Susanoo isn't going to last (hopefully). Eventually, Kakashi will run out of Obito's chakra or Obito will return to the spiritual realm (it's still not clear if he returned). If Kakashi's use of Susanoo is only temporary (hopefully) then this can be forgiven, eventually.
Some inconsistencies however
1. Amaterasu + Tsukuyomi = Susanoo
Kishi wrote in a databook that Susanoo is awakened by the combination of Tsukuyomi and Amaterasu. That is clearly not the case anymore. If two any techniques are required, than Sasuke has Amaterasu, Kagetsuchi and presumably Tsukuyomi as that was transcribed to him by Itachi's seal, apparently. Madara has Tsukuyomi and likely Amaterasu as well (I'm assuming this based on the fact Tobirama seemed to know of Amaterasu and likely experienced its use by Madara). Obito's eyes only have technique however, so even if any two powers would be fine, this is an inconsistency.
2. Chakra
Kakashi should not have the reserves to pull of a full fledged Susanoo. It simply doesn't make sense. Even if Naruto gave him chakra when he healed his eyes, I doubt Kakashi has it in him to pull it off, but apparently he does now. One could point to his more frequent use of Kamui to illustrate greater chakra reserves, but I would remain sceptical given the size and power of a PS.
3. "Perfect Copy"
Even if Kakashi could copy other techniques of the the same KKG, how would he have gone about that considering Sasuke doesn't use any hand seals for the jutsu, and even if Kakashi could perfectly read the ways in which chakra was manipulated it would require some ridiculous level of precision and attention to detail. It just doesn't feel right across the board.