Though Bear is a relatively recent name, it has ancestors that can be traced to Medieval Europe to the point that Bear just seems like a naturally modern and legitimate alternative.
Bero & Pero are the Old Saxon progenitors that can be traced to 9th-century Austria. There is the Anglo-Saxon Beorn, German Bern, Dutch Beren, Norse Björn and the Yiddish Ber.
Ber was used by European-Jews for centuries. Ber spun off the Hebrew name of Dov (bear) and Ber-Dov was a popular compound name found among Eastern-European Jews up until the mid 20th-century. It spawned diminutive offshoots such as Berl (not to be confused with Beryl) and Berek, borne by Berek Joselowicz, a Polish general of Jewish extraction who participated in the Kociuszko Uprising and was the first commander of a Jewish regiment in modern military history (1764-1809).
There are numerous Germanic names that have the element of Bern, the most notable being Bernhard or Bernard. The Swiss city of Bern is of the same etymology, and while folk etymology does link Berlin with bär (bear), it is actually related to a Slavic source meaning “swamp.”
The root of the word bear itself and its other Germanic cousins, all relate back to a Proto-Indo European euphemism meaning, “the brown one,” which relates back to the name Bruno. It is suspected that the ancient Germanic peoples feared the bear so much that they had to use a euphemisim to describe it in fear of conjuring its power if the animal’s true name was evoqued. The bear was a sacred animal among the early Germanic tribes that they also used its euphemistic name on their children.
As a given-name, Bear has recently entered the Top 500 Most Popular Male Names in England & Wales, currently ranking in at #384 (2018).
A notable bearer is Bear Grylls, borne Edward Michael Grylls (b.1974).
Feminine forms are Bera & Berna.