Fact Checked Speculation > Memory Backed Speculation.
Shizune is Dan's (Tsunade's BF) niece. Dan's sister (probably Shizune's mother) died in the war (and he was possibly taking care of her after this). When Dan later died and Tsunade left the village she took Shizune with her as her apprentice, they traveled together until Tsunade came back to the villiage to become Hokage.
I don't think it's all that strange for an orphan to work for the person who looked after them, raised them, and trained them after their family was killed. Such a person might even feel a debt of honor to that person for taking care of them and thus put up with their bullshit attitude out of gratitude for what they did *cough* Haku *cough*.
But I guess that's exactly what a tool isn't it? A person willing to give up their own desires and aspirations in order to server the desires of the person who was there for them at a miserable time in their life and took care of them.