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As we know, Usagi is based on the real-world ronin and 17th century duelist Miyamoto Musashi. If Usagi's shogun were also to be based on the actual historical personage of the time, that would the first Tokugawa shogun, Ieyasu.
In the real-world history, Miyamoto Musashi and Tokugawa Ieyasu were on opposing sides at the Battle of Sekigahara. Tokugawa won and became shogun, leaving Miyamoto a ronin. If the same or similar events and relationships carry over into Usagi's world, then the shogun would be the direct cause of Usagi's status as ronin and it would be unlikely that Usagi would be happy about that.
However, as to what Sakai-sama has done / will do remains to be seen (at least for me.) For all I know, the shogun in Usagi's world is a platypus named Steve.
How very noble
He who finds no satori
In the lightning flash
-Basho
I can only assume the Shogun in UY is roughly similar to the Shogun at the time.
Plus the Shogun is mainly known just in name, usually by some people who work for him mention stuff like "Shogun wanted [something]" and thats it. He isnt mentioned that often.
Stan has mentioned previously that the events in Usagi's universe fit in (albeit fictionally) with historical Japan as it was in the early 1600s. The Shogun at that time was Tokugawa Ieyasu, as pointed out. The Shogun's actual name has never been revealed in Usagi's universe, but Stan has always based the more notable characters on actual people (Miyamoto Usagi = Miyamoto Musashi; Lord Hikiji = Date Masamune, etc). You can bet that both the Shogun and Emperor are based on the actual Shogun and Emperor or that time period.