@Lord_Scorpion34 I recon her rat companion is a lab rat she picked up somewhere, but has a deeper resemblance with Ash's relationship with the other legends. Her lab testers' background compels her to torture the rat with various trails and attempt to teach it obedience (her emote where she places cheese on the end of her sword and instructs the rat to eat it for instance), but in doing so, the rat has become a companion of sorts, providing ash with the strangest of company while she works in her roguish way.
It's a similar relationship with her and the other legends. Through the Areas and her various voice lines, Ash is testing the legends and attempting to teach them in the ways of combat for her own curious amusement. But she is showing early signs of becoming sentimental to some of them, especially Path in the latest lore clip released. Though she is hiding it from herself and the legends by attempting to keep the relationship with her lab rats strictly professional, as she still believes anything else is a form of weakness existing in her past self.
Thank you for listening to my TED talk
