Workplaces often have employees of varying ages and there's nothing wrong with older employees having mentorship relationships with younger employees, but this isn't that. You cannot, and should not, have an individual "friendship" with a female intern that is based around your inappropriate disclosure of your personal life.

There was no need to ask a female intern half your age about issues pertaining to your ex. Even if this weren't a much younger female intern, bringing highly personal and complicated issues into the workplace in general is a breach of professional boundaries.

The intern has attached herself to you and now views you as a father figure.

Apologize for involving her in your personal life and reiterate that while you are sorry she has experienced such instability, your relationship needs to be strictly professional from here on out.

If you have any direct supervision over the intern, ask your boss to be reassigned to another intern or have other adjustments made. It will only harm her further if she continues to work with someone she's developed an unhealthy attachment to.