Prior to the reboot, he was active for like 15-20 years, so it's easy to believe that anything he didn't know right at the beginning of his career, he learned over the course of it. Realistically, several fighting styles seem to overlap in terms of form and technique, so it's not that much of a stretch to think that Batman became skilled in one and glanced at some other related/similar arts and gleaned some moves to incorporate into his style. In regards to his smarts, that could be attributed to being wealthy and fortunate enough to afford the best schooling, so when he wasn't physically training as a teenager, he was book learning.
Not to mention, Bruce learned via unorthodox means. Most people, when they train and learn martial arts, they have downtime. They do things to relax and have families that they go to spend time with and friends they associate with; Bruce went off the grid and pretty much trained non-stop for years with people who didn't necessarily train the way most masters would train a pupil. Not unreasonable to believe that under such extreme training regimen, someone could master a fighting style relatively quickly.
But he also lives in a universe with beings who defy physics, and that includes normal humans who're aren't nearly as smart and aren't as well versed in as many fighting styles as Batman. The rules of his universe shows that someone who spends the right amount of time and studies under the right people can become polygots, polymaths, and become one of the best melee fighters on the planet.
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