Obi-Wan saw Anikin as a brother, he told him he was like his brother when they fought. No Minato didn't see him anything more as a student; a teacher usually view like 1 student as a brother or son but not all had this. Ex: Jiraiya voiced that he thought of Minato like a son, Gai treats Lee as a son and Lee even copied Gai's look, Hiruzen favored Orochimaru when he was younger, Tsunade probably view Shizune and Sakura as young sisters.
Minato seems to have spent more time with Kakashi so I see Kakashi more like Minato's younger brother than Obito. Minato even taught him Rasengan but Kakashi couldn't add lightning to it so made Chidori. Kakashi also was Minato's 2nd hand man as leader if the Anbu when Minato became Hokage and trusted him with the task of watching out for Kushina for months. Obito still respected Minato as a good man and teacher and Minato treasured all of his students.