The bear resides in Cyro, in a nice peaceful spot. You cannot kill the NPC, though I don't think you should even try, due to the possible circumstances.
It is supposedly a memorial of a player that passed away from cancer who was a fan of the elder scroll series and bears. And as the story goes, Zos and Bethesda decided to have NPC bear named after them, which is Lin. If you look at the NPC data you will only see question marks.
Here is a video that someone took of Lin the bear and a link to the NPC info.
https://m.youtube.com/watch?v=I1wIGjKMbGchttps://elderscrollsonline.wiki.fextralife.com/Lin
Hope this answers your question. Some say it is part of a quest others say a memorial. Though the bear doesn't die, so I would be inclined to say it is a kind hearted gesture by Zos for a player that has passed on. Since I doubt Zos will confirm this because of the family's privacy I don't expect an answer beyond what I was told. If it is a quest then I am sure another player can prove this, so you have an answer for your curiosity.
Edited by Girl_Number8 on March 12, 2019 3:06PM