I’m vaguely aware of the story of the invading Mongol army being destroyed by a storm, but was wondering what inspiration from history the game story may be grounded in... if any.
I will pretty much cite myself (with slight alterations) from an earlier question on how much the game is historical here, since I am referring to multiple posts by other authors.
That being said:
There have been several questions (with answers) on the Mongol Invasions of Japan in the past, for example here , here , here , and here.
Furthermore, related to the game's historical accuracy, you can also find this and this thread.
(note: answers in these linked threads have been given by u/ParallelPain , u/krishaperkins , u/wotan_weevil , and u/Erina_sama )
To summarize, yes Yuan China, which was under Mongol rule at the time, indeed launched two large-scale naval invasions in 1274 and 1281, and retreated both times. In both cases, typhoons were given as the reason for their "defeat" (although that is debatable). The island of Tsushima, which is closer to the Korean peninsula than to Japan proper, was attacked first and overrun (there were only about 80 warriors on the island, who all were slaughtered pretty much instantly).
Although the team did travel to Tsushima and took inspiration from the real-world island for their location design (for example, the castle actually existed), the characters and the plot are entirely fictional. The clothing, weapons and armor are mostly ahistorical, stemming from later historical periods. (I do find it especially amusing how fancy the clothes are that some of the peasants wear.)
The whole "bushido" and "samurai" blabla is hardly historical, since neither did warriors at the time self-identify as "samurai" - which is a development of the later Edo-period, when the warrior elite became socially locked into being a samurai "caste" - but rather as warriors, nor did the whole code-of-honor-"bushido" as per popular imagination exist in this form, since that is a fictional creation constructed in the late 19th/early 20th century and loosely based on, mostly, Edo-period writings.
For more information regarding these and other things, check the other questions and their respective answers :)