No, but the fear needs to be irrational to be regarded as a phobia. Being black widow-phobic wouldn’t make sense, because fear of black widows is entirely rational. But arachnophobia implies an irrational fear of all spiders, and getting panicked by a daddy long legs isn’t really that rational. However, apparently fear of spiders is innate in humans because they don’t appear to be at all threatening but obviously very much are so there’s a theory that we are born with an instinctive fear of them as an evolutionary development.