He married into a clan that gave respect to his clan name. The Uzumaki admired the Senju for their feats and that's why people seem to think that they had kids. But it is never stated by Hashirama or Stunade that she is Senju only Madara which how would he know if he wasn't around or dead.
Just blindly saying she is Senju because it's stated by others doesn't make it so....an like i said her nor Hashirama stated that she was Senju, just that she's his granddaughter. So for me personally i won't call her Senju Stunade, but just sanin Stunade.
Where do I say that she is Senju only?
She says herself that she is Hashirama's granddaughter.
Hashirama says it. That she was his first grandchild.
She is even called princess Tsunade due to her heritage.
She talked how she didnt inherit the mokuton.
She is a descendant of Hashirama and Mito.
Her grandparents are Senju and Uzumaki.
Her parents are unknown.
What do you want to say than?
That only Mito is her biological grandmother and that Mito had a child with another man before/after Hashirama?