Fabricated family pedigree
First of all, do you know the difference between Sei(姓)and Myuji(苗字)? The meaning is different though both are translated into the family name.
Samurai's Sei(姓)is either 源(Minamoto),平(Taira),藤原(Fujiwara),橘(Tachibana), or 菅原(Sugawara)what the ancestor was given from the Imperial court.
Each Matsudaira, Tokugawa, and Nitta is Myuji(苗字)that shows the name of a place where they live.
When Ieyasu was given 従五位下三河守(the official rank title for the feudal lord in Mikawa) was gotten from the emperor, Ieyasu changed his Sei from Matsudaira into Tokugawa.
Shogun Ashikaga was a genji at that time. Genji is a member of the Minamoto clan. For almost lords at that time such as Nitta, Takeda, Imagawa, Kira, and Hosokawa, they were relatives of Ashikaga clan.
Matsudaira is not Genji but Fujiwara clan.
Ieyasu falsified the family pedigree when promoted to the lord, and insisted that he was Genji and Ashikaga's relative. He insisted that his ancestor was Tokugawa who is relative of Nitta clan. He insisted that Tokugawa was a relative of Ashikaga because Nitta was a relative of Ashikaga.
Tokugawa is a name of a place in Kyoto (the capital city) ,but Matsudaira is a name of a local place.
Though the court did not admit his insistence at that time, his insistence was admitted when he ascended the Shogun in 1603.
Mai Shikata
If you want to strike your opponent, you should let him strike at you.
If your opponent strikes at you, he himself will already have been struck.
-- Yagyu Munenori