The most successful modern storylines avoid the ick by ensuring the "animal" is either a full human shapeshifter or a clearly allegorical creature (like a god in beast form). The moment the animal cannot verbally consent, the "romance" becomes a horror story.
The most traditional version of this relationship is the one we see in The Shape of Water , King Kong , or even The Last Unicorn (depending on your interpretation). Here, the "animal" is usually a masculine-coded creature—powerful, misunderstood, and dangerous. The female lead acts as the conduit of civilization and empathy.
However, in Ovid’s Metamorphoses , we find more sympathetic turns. The tale of (Echo being a nymph who fades into a disembodied voice) touches on unrequited love for the non-human. But the most direct example is Pasiphaë —though she is the female human who lusts for a bull, the inverse (man lusting for animal-woman) is often censored. When it appears, it is almost always punitive.
Example: Beauty and the Beast — Belle sees the Beast’s humanity before the curse breaks.