It involves a radical life change, labels many existing friends and family members as negative influces, and replaces a lot of day to day life with other activities (most of which are exclusively straight edge). Plenty of people live like straight edge people without the label, the clothes, the music, the circle of friends, etc.
Straight edge is not a cult. It is just a way to live. If you don't want to drink or smoke, then don't. It's completely your choice but it doesn't make you part of a cult. They don't have gatherings. They don't hurt anything. Also, by definition it can not be a cult. It is not a religion. It's like saying the ten commandments were cult letters.
Cults generally involve some kind of faith or religious belief, but straight edge, or the choice not to take any intoxicants, is simply a lifestyle choice. Just as people should be free to consume what they choose, people should be free to refrain from consuming what they feel is harmful, and that does not mean they are in a cult or should have any negative association with that choice.