It (rather, he) followed me through the doors as I tried (to the best of my ability) close each subsequent door before it... he... had the chance to make it through. The escape through each room was a chore; a game; a routine. Something obtained for the purpose of comfort had given that and more, but at the sacrifice of privacy, freedom, and a self-learned love of loneliness. The grass is always greener -- always greener.
I like how both you AND the kitty long for the greener grass. I think you could reinforce the idea that the kitty is trying to explore the other side rather than just wanting to follow.
Given your desire to find privacy, it would make sense to always refer to the kitty as “it”, perhaps even pointing out your respect for “its” privacy (if you can afford the words).
While I believe the theme of this story is both prominent and the point, I wonder if some more concrete imagery would make this story come alive for me. To see the kitchen, understand the geography, etc. Nitpicky: "close" either needs a "to be" verb with it or, if the story was made into present-tense, "closing."