Via Twitter, developer Omega Force and publisher Koei Tecmo announced that the demo for Samurai Warriors 5 is available on PS4, Xbox One, and Nintendo Switch in Japan. As specified in the tweet, the trial version features the beginning part of the story. Also, save data can be carried over to the full game when it officially launches on July 27.
At the moment, we are not sure when/if the demo will be released in other regions, but if you happen to have a Japanese account of any of the above-mentioned platforms, you could give it a try.
Otherwise, our preview text for Samurai Warriors 5 written back in January can be found here if you'd like to know more about the game.