This is How Did I Become the Princess?, aka Starting From Today, I'm a Princess?!, a Korean manhwa from 2021.
Once upon a time, I was a weary, nameless warrior with neither family nor money. Some might say this was just my lot in life, but not me. When I am given the chance to return to the past, I take it, hoping to return to simpler times. However, after turning back time, I am shocked to learn that I am actually Princess Nanael, the long lost daughter of the emperor of Ovyulia. With my new name and new home, I’m dead-set on not losing my family ever again.
The main character is a hero that defeats the demon lord, and is subsequently poisoned by nobles that feared she'd betray them, so they try to get rid of her first. She contacts a dragon she met earlier and it grants her wish of turning back time. Upon re-awakening she discovers she's the emperor's daughter.
The dragon does stick around as a cat-like thing.