Why couldn't Kakashi pass through things? After Kakashi studied Tobi's movements during the Fourth Great Ninja War, he realized that his ability to teleport and pass through things is all one single jutsu, he even said it himself. The ability to pass through things is when Tobi sends a specific part of his body to the Kamui Time-Space. When he teleports, he takes his entire body with him. Even being said, Kakashi couldn't even pass through things until his fight with Kaguya Otsutsuki.
Knowing this, shouldn't Kakashi be able to pass through things as well?
Note that the two eyes of Obito performs two different actions. The right eye performs the space-time ninjutsu or the teleportation jutsu for the user only (so it a short range version of kamui). The left eye performs the teleportation jutsu for the enemy that is targeted i.e. the user looks at the target and concentrates so that the target is transported to the space-time dimension (so it is a long range version of Kamui). Kakashi has the left eye, so he can perform the long version of Kamui while Obito performs the short version of Kamui.
Even if Kakashi got the right eye, he couldn't be able to perform such a chakra-inducing jutsu because he is not an Uchiha. Note that Kamui is the speciality of Obito's Mangekyo Sharingan (which is Kekkei Genkai exclusively for Uchiha clansmen only). So, using Kamui demands a great deal of chakra for him to perform. It takes him an extended period of time to gather enough chakra to use it, and once he does he cannot use it again. Even if he does, it won't be as effective as he would have wanted. Also, repeated use of Mangekyo Sharingan's abilities can damages user's vision, eventually reaching the point where it causes the eyes to bleed. For Kakashi, it would be very, very painful.