They existed, but I wouldn't call them common- I doubt there were more dueling piano bars then than there are today.

Dueling piano music came into existence around the 1890s. A number of factors contributed to the entertainment never becoming wildly popular-

1. Startup cost- To invest in the purchase of 2 serviceable pianos would have been more expensive than most other forms of barroom entertainment.

2. You would need not 1 but 2 very talented pianists. Anywhere other than a large city, that might be difficult or impossible to find. (I'm not just talking "good enough to be a professional," to make that style effective you really need the best of the best)

3. The bottom line is, to play at a high level with another person requires rehearsal. You might see two pianists playing and acting as if they are showing each other up, but in reality its part of the show.

Now, if you skip ahead a few years to the 1920s, well, then musicians "duels" (called "cutting contests" by players) did happen for the express purpose of showing each other up. It began on pianos, but rather than the duel in your clip, it would have looked more like this. (If you watch the clip, please be aware that Jelly Roll Morton did not invent Jazz as the movie says- artistic license is being taken by the script writers there-I also doubt that someone would have chosen to play the exact same song that had just been played, but you get the general idea.)

This continued into other instruments during the Jazz age- when the vehicle of performance went from stride piano playing to combos or big bands with parts expressly designed to improvise over, anyone on any instrument could get up and try to prove that they were the best.

The heyday came in New York in the 1940s when bebop was developed and the emphasis of jazz started to stray more towards improvisation over chords rather than the melody of the song. Charlie Parker and John Coltrane (both saxophone players) were two of the undisputed best at making someone feel like they had been gifted a new sphincter during these sessions.

(there's apparently even a "Charlie Parker Cutting Contest" in Kansas City named after Parker)