Tenorio is pretty dead-on when he makes his claim about Ultima's Owl right before the climax of the book: "'It is the owl! Do you hear, little bastard! It is the owl that is the spirit of the old witch'" (22.430-432).
Captain Obvious Tenorio tells us straight up that the owl symbolizes the spirit or soul of Ultima. And he's right. When the owl is killed, Ultima dies shortly after, and Antonio realizes that when he's buried the owl, he's also buried Ultima. Kthanksbye. We're done here, right?
Not so fast. It's a bit more complicated than that. The owl also serves to demonstrate Ultima's violent and vengeful side. While we always see Ultima as a healer and a wise woman, the owl rips out Tenorio's eye. That's some serious business right there. If the owl symbolizes Ultima, then in a very real way, she's the one ripping the eye out. She's not all hearts and flowers and blessings and exorcisms, no sir.
The owl isn't just about lashing out, though. After all, Ultima's got many layers to her onion, and so must the owl. When it arrives along with Ultima, Antonio is frightened of it at first. However, he soon comes to think that "Its song seemed to say that it had come to watch over us" (1.454-455). Like Ultima, the owl is a protector, and in that sense, Antonio's relationship with the owl parallels his relationship with his spiritual mentor.
The owl ain't all about Ultima, though, to be sure. There are bigger things in play that one woman's soul, like the deep connection between humans and nature. In his descriptions, Anaya makes that much clear. The owl's song, for example, "calmed the moonlit hills and lulled us to sleep" (1.453-454). When nature calms down, man rests easy. And apparently the owl can make that happen.
This connection looms large throughout the book, and it's an understanding that Ultima passes on to Antonio. But in the owl, the connection becomes almost literal, as it's an animal that shares the very spirit—the very soul—of a human being.