At one point Penguin yells "wind - forest - fire - bird", which is a twist on Takeda Shingen's personal motto "wind - forest - fire - mountain"). This was a reference to a Chinese text on warfare: "Be as swift as the wind, silent as the forest, fierce as fire, and as immovable as a mountain".
Bats are actually seen as symbols of good fortune in China (and therefore feudal Japan/other countries with Chinese influences), due to the similarity in writing/pronunciation between the words "bat" and "luck".
When Catwoman first meets up with Batman in the feudal age, the hand puppet she uses to get his attention resembles a black version of a "maneki-neko"; the traditional (white) cat statues that raise or wave their paw to bring in good fortune.
In advance of the physical release of Batman Ninja on U.S. home video on May 8, 2018, Warner Bros. Home Entertainment arranged for a special two-night event in New York City that included special screenings of Batman Ninja on May 1 & 2.
Poison Ivy's two pet monsters both carry a Chinese character on their heads: one means "poison", the other is the first of character of a Buddhist deity's name, as used by Uesugi Kenshin on his banners.
First when Grodd Combines the Mega Machines then Later on The Joker Reforms it to a Giant Fighting Robot Could be A reference to The Super Sentai or Mighty Morphing Power Rangers for Americans, where they start off with the Tank Mode and then Reform it to the Megazord Mode on the Show's First Season