I'm not sure, most people probably wonder this. But he brought Rin back to life so I'm assuming he feels like a parent or guardian for her since all her family members died plus its like a father daughter connection considering he brought her to life as a mother would to a child.Also when Rin was living she care for him when he was injured in the forest but Sesshomaru didnt need any help from her because hes a demon and doesnt eat human food, guess deep down he appreciated it but didnt show it.
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he has grown attached to her (he won't admit it of course). he thinks of her as a daughter-like figure i guess. or maybe something for him to protect.
WARNING; SPOILER BELOW;
in the end seshomaru leaves rin in the village to train under lady kaede but he often comes to visit her and gives her gifts like new kimono's, jewelery, etc. she totally has him wrapped around her little finger.
because rin was so kind to sesshomaru when he was down and i guess he grew caring feelings for the little girl
no rin leaves sesshomaru in manga near the end but not in the anime
sesshomaru greatest desire is rin
No he didn't name her. Before Sesshomaru found Rin, she had a family, but they were killed by a band of thieves. When Sesshomaru found her, she was dead, but he brought her back to life :)
Lord Sesshomaru never shows his feeling towards Rin, but he would never hurt Rin because he cares for her to much.
yes people can like Rin and Sesshomaru as favorite characters n Inuyasha! everyone has an opinion:3