Always ask before you bring anybody with you, and ask when you book rather than on the day. A message on the booking thread is enough.
The answer is often no, and it is usually not personal. Many studios are a single room or a rented suite with one chair and no waiting area. Some carry insurance that does not cover people who are not being treated. Chemical services involve products that should not be sat next to for two hours, and a colour appointment can run three.
Where children are concerned, expect a firmer answer. A child who cannot be left unattended in a room with hot tools and chemicals puts the studio in an impossible position, and the honest studios say so up front.
If you cannot come alone, say that early. A studio that cannot accommodate a guest may still be able to move you to a quieter slot, and one that shares a larger space may have somewhere for somebody to sit. That conversation goes much better a week ahead than at the door.