If the cat has become literally stumbling block, because if he started fighting, scandals - pair better hurry to leave, not tormenting each other. But if they value relationships, we must try to find some mutually acceptable compromise.
The girl needs to understand that love for animals - that's fine, decent quality, but you can not turn it into an obsession. Everything is good in moderation. Alas, some animal lovers this measure itself does not know clearly dividing humanity into "brothers in spirit" and "idiots who hate all living things." Extreme, bordering on fanaticism, harmful in any case. Nobody forbids a girl to love your cat, but can not impose this love to others. It is enough if it is a young person does not show hostility to the animal. It is not necessary to require that guy all the time stroking the cat, he took him in his arms, saying that while he certainly understands what good is an animal, and love! Will only get worse.
Guy, respectively, should be attributed to the feelings of a girl with respect and condescension (of course, in a good way). In no case can not show hostility to the cat, making fun of her. Like, what good is it found in the hairballs, and also bad smells from cats. You can overpower yourself and pet the animals, scratch behind the ears. This will be enough to start. And to take him in her arms, the more false admire its beauty and grace is not necessary. It is possible that over time, the guy really gets used to the cat, and even love. But it has to happen by itself, without any pressure from the girl.