The problem is not so much their hair but more the saliva they put on it. But after all, what difference does it make? For the nose and bronchial tubes, it's the same: we cough, our nose runs, we're merely allergic.
Some families are forced to separate from their favourite animal and that's very sad. Before reaching this disappointing end for everyone, consider welcoming an animal, especially if you (or one of your children) are already allergic.
However, if you are a pet owner, a few right daily actions can mostly limit the damage.