At the beginning, Nagatsuki-san did play the analyst role correctly. He asked the uncomfortable questions to help Akito-san challenge his way of thinking. After that, things became muddled.
He knew that Akito-san (his client) had an unnamed, complex, exploratory relationship with Miharu-san (I believe he saw here picture on Akito's phone). Yet, when he met her by chance, he decided to court her while knowing his client had "something" going on with her. Akito's relationship with Oshiko-san has nothing to do with this; a therapist should never get involved in their clients' lives, it's unethical.
I thought that Oshiko-san had a different therapist. When she took Akito-san to the clinic, she stated she was being treated by someone else. Now, seeing Oshiko-san contacting Nagatsuki-san for a chat in the middle of the night is a major red flag. If he is indeed Oshiko's therapist, he is breaking his clients' confidentiality by sharing information with them. Being together with Mahiru-san makes this worse due to his ability to manipulate the three people at his will.
And no: therapists do not have psychic abilities to know what you are thinking. That is why they ask so many uncomfortable questions to get the answer out of you and making you cry in the process.