"My employer has a strict policy on social media, I'm not allowed any presence at all - professional or personal. Can they really enforce that?"

Your employer is certainly in the minority in having such a strict policy. Although most firms have policies restricting employees from using social media excessively during working hours and requiring them to take care to not damage the business' reputation or customer relationships, it's very rare for an employer to ban staff from having a personal online presence at all.