The name, "Bruin".
And is it reasonable to assume the first had some influence on the naming of the second?
(Or is there a Bruin sports team that predates both?)
According to the Hockey Hall of Fame, the Boston Bruins began play in 1924.
According to the UCLA Alumni Association (click Mascot), UCLA adopted the Bruin as their mascot in 1926, having been previously known as the Cubs (1919) and the Grizzlies (1924). The Cubs were selected as the original mascot because the university was orginally the Southern Branch of UC Berkeley, who were known as the Bears. (The bear was well established as a symbol of the state of California and appears on the state flag.) They apparently switched to Grizzlies to try to get from under the shadow of Berkeley while retaining the "bear" imagery. In 1926, when UCLA joined the Pacific Coast Conference, the University of Montana was already the Grizzlies, so UCLA changed to the Bruins - yet another word for "bear".
It's possible they were influenced by the name of the Boston hockey team, but it also could be simply a logical progression. I don't think the term "bruin" for a bear (coming from Dutch, I believe) is so obscure as to make it improbable that two organizations could independently select it.