But the Yin seal is just a general name. Its not the seal's real name. The seal is called Byakugou as it was revealed by Katsuyu.
Anyways, care to explain why would Sakura have a different seal?
Where did you got it that it's a general name? And where did Katsuyu revealed it's called Byakugou in regards to Tsunade? Katsuyu only made a reference at Sakura's seal, not Tsunade's.
And why not have a different seal? Not to mention, Yin is associated with dark, so let's say that that is Tsunade's proficiency when it comes to Yin chakra. And Sakura's one meaning White, could be associated with Yang chakra.
Also, Sakura is associated with White and Green Tara, which is known as the female counterpart of Kannon. And funny enough, Hashirama's technique, one of them, is translated as Senju Kannon. The Kannon being mostly known as being a female. Anyway, I've explained it more
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So as I said, there are all the reasons Sakura could have a different one. Don't forget we don't know which seal Mito possessed. But the difference in names is clear. One is Infuuin which belongs to Tsunade and one is Byakugo no In which belongs to Sakura.
We could link them, however, in the case in which Kishi decided that Yin seal could be upgraded to Byakugo seal, just like Sozo Saisei is upgraded to Byakugo no Jutsu.
Otherwise, besides the design, at the moment at least, they seem to have nothing in common.