I still don't get why they're fighting.
shitty childhood as a result of superstar popularity during childhood that led to being trafficked and trained as a highly intelligent assassin. The idea is the best weapon is a smart and loyal one but can only be forged starting in childhood since children are extremely easy to manipulate.
--> Maria aka Shue/Xue aka Yuki accepts killing as a method of making a living in addition to whatever other brainwashing she took in --> Anna's personal ideology conflicts with that and cannot be reconciled --> Anna leaves and breaks apart their family bond, they both remain this all the way into adulthood --> Yuki somehow becomes connected to the Yokoya Family (Hityoshi's family) at some point + the brainwashing seems to have subsided which will eventually be explained leading to our present day bonds. Anna lives for all the baggage + revenge.
Let's remember that when Yuki first arrived, she said she had a referral from her master to become a servant for the Yokoya Family. Naturally Hitoyoshi, never taking whatever his father said seriously, brushed it off and didn't believe it. This is why when his father returned home, he felt left out and surprised that Yuki and his father recognized each other. Ofcourse, Yuki heard of him from her previous master but only met him for the first time at the store.
SUPER SIMPLE SUMMARY: both traumatized --> Maria aka Xue/Shue aka Yuki is trying to move on now and is advancing. Anna is still stuck in her trauma, unable to or not ready to move on and this causes a conflict between the sisters.
If you still don't get it, try thinking of soldiers returning from combat duty that need time to get over PTSD. Many get through it, many need more time, help or cant get through it.