You have asked a very interesting question because at first glance, these two stories appear very different. Twain writes his story with characteristic understated humour, combining use of hyperbole with a dry, dead-pan tone to increase the comedy. Brose's story, on the, other hand, is set during the Civil War and features a man being hung on a bridge. Two very different themes, I am sure you will agree!
However, the similarity lies in the period in which both stories were written and the school of literature which is used to describe works produced during this time. Both are examples of realist fiction, in that both reject the larger than life hero of Romantic literature, they give detailed depiction of ordinary characters and realistic events and scenarios (even if Twain's story isn't realistic, the setting definitely is), there is an emphasis on characters from the lower classes and there is an avoidance of exotic and overly dramatic events (except of course in the Tall Tale in Twain's story). Lastly, another key aspect of Realistic fiction is the use of everyday speech patterns to reveal class distinctions. Twain in particular is a master of this and uses the vernacular extensively in his writings.
In addition to the most salient of points, that both stories, "An Occurrence at Owl Bridge" by Ambrose Bierce and "The Celebrated Jumping Frog of Calaveras County" by Mark Twain fall into the same genre of Realism, there are a couple of minor similarities between them. One similarity is in the altering of points of view within the narrative. In Bierce's story, for instance, there are three variations: omniscient,in which the narrator seems to know everythings, objective in which the narrator reports without comment, and third-person limited in which the narrator zooms in on the thoughts and feeling of a single character. Thus, the story is divided into three parts, one of which reads somewhat like a tall tale. In Twain's story, the tale told by Simon Wheeler rings of an exaggerated tale; moreover, the narrator certainly contrasts with the anonymous one, as do the third-person limited and objective contrast with the omniscient.