あなた can't do anything to get them. I don't know what anyone told あなた but it seems there is no way to get any animal other than farm 動物 (like chickens and cows), a dog, cat, and a horse. If あなた befriend Lumina's grandmother, あなた can get one of her ネコ eventually; and at some point Takakura will bring あなた a horse. Other than that there are no other animals.
For instance, the little animal sometimes near the dig site with a bandanna around its neck. あなた can't keep it, I'm pretty sure. I've tried so many ways. I can keep checking though.
Edit: Okay, I found a really trustable site/person (
link) and you'll see it says あなた will never recieve that チワワ (the bandanna animal). And it doesn't talk about the others. I assume they'd know if it was true. You'll just have to assume that あなた can't get any other animals.
