Being Eliza's pet does affect a few interactions, yes (although you'll be free to do your own thing for the most part). There is an in-game way of getting her out of your life: it involves finding a hidden dragon who can grant you 3 wishes. You can search the walkthrough if you want to learn more about him (make sure to read his section entirely, as you can get an optional bad end after the third wish, depending on how you acted around him). Also, if you are a cowgirl for the Zanti farmi, the problem will "fix itself" (and the Zanti will get a new cow).