Sounds like displaced anger/frustration to me, and the poor puppy is the "outlet" for your daughter's pent up whatever.
If she cannot develop "empathy" or responsible handling of the pet however, then she should not be allowed to be with the pet. Or the pet will become unfairly affected.
Does your daughter's eldest sibling boss her around or do teasing toward her or anything that is not age appropriate??? If so, your daughter may be just striking out at the dog, for whatever is done to her... out of frustration, because the puppy is "harmless" and cannot lash out at her like "older" humans can and is sort of defenseless.
Your daughter is already 5 years old... at least in Kindergarten? So she 'knows' in her heart what is right and wrong. It probably is true, that she does not know "why" she is hurtful and abusive to the puppy... all she knows is that she can 'pick' on it and it relieves her of stress perhaps.
In some way, she is doing to the dog, what she feels inside maybe... and it is an outlet for her.
I would talk with her, or perhaps ask a Teacher on how you can help her to express herself or how the ENTIRE family can help her...
Perhaps even ask your eldest child, what she thinks? I know, as a child and being the youngest, my sister was SUPER bossy toward me and teased me a lot... it pissed me off, and I didn't understand why "I" got scolded for things... it was probably because my sister was older and more able to defend herself and verbalize things and get herself out of situations. Versus for me, being the youngest, I was not as quick mouthed as my sister... thus, I got blamed more.
But certainly, the puppy has to be helped too... or it will become afraid of your daughter.
All the best,