Khan is a catalyst in Kaira's life, and since she was the protagonist of the story, we sadly are not shown much about him in the movie.
Khan values the comfort of his patients, and he is willing to step outside the comfort zone for that. He conducts his sessions at his cozy and welcoming home, and at times also outdoors. He spends his first day telling Kaira a moral story that she relates to, thereby assuring her that he is capable of understanding her problem.
Even when it comes to his own self, he values comfort. We see him dressed like a Silicon Valley techie at a very formal event. He probably gave himself the nickname 'Jugs' because he realized that 'Jehangir Khan' sounds serious and intimidating, while his patients will relate to Jugs better.
Khan uses humor to normalize the heaviness of the fact that one is seeing a shrink.
Khan also does the job of keeping the entire focus on Kaira beautifully. He shares his own stories only to further the sessions (like talking about his son and divorce to help Kaira ease-in to discuss familial issues)
When Kaira confesses her feelings to him, we all could feel the suppressed reciprocation in Khan. He shouldn't overstep his professional boundaries, and he doesn't (he isn’t present even at her movie screening)
Jugs is a therapist, so a big part of his job is to listen neutrally. But there are times when he is unable to suppress his emotions around Kaira, and that is when he turns to his eye-drops. We see him use the drops twice, once when Kaira is breaking down and sharing her childhood trauma, and the second time during their last session. I feel he hides his eyes behind those drops to not let them convey his emotions for her.
The entire point of starting therapy is to eventually not require therapy. And Khan understands this. Getting therapy is similar to learning to ride a bicycle for the first time. You are wobbly and unstable, so someone is holding the bike to help you balance. But you cannot have someone holding your bike forever. The whole point of it is so that you are capable of riding the bike on your own, without any training wheels, and that is what Jugs does for Kaira.