The brown bear is probably the most bear of all bears. At least if you judge by the name. Ursus arctos, it's binomial (or Latin) name is literally bear bear from Latin ursus meaning bear and arctos being bear in Greek.
It would also sound more suitable to call the polar bear which such an obvious reference to the arctic, but no, the polar bear is just ursus maritime. But the arctic is actually closely related to bears, just not the polar kind. The name arctic is indeed derived from the same Greek arctos. But instead of referring to actual bears (polar or not) the name is a reference to the constellation ursa major (or according to some sources ursa minor) a.k.a. the Great Bear which can only be seen in the northern sky.
So arctic is the region where you can see the "bear constellations", but it's actually also the only polar region where you can see the actual bears (polar or not). And the Antarctic literally meaning "no bears" in that sense is just a coincidence as the people living during the time these regions were named hadn't ever visited the southern polar areas to know whether there were bears or not.