NO..you do NOT have to supply them with your social # or her's for that fact. Although they probably already have hers.
The only time you have to supply your social security number is when
You are dealing with a bank
Asking for credit.
Talking to a government office, such as social security or social services.
Do NOT give them your social security number. Tell them you have an individual policy against releasing this information. They may want to argue the point, don't give in. They don't even need it to file a claim on your health insurance. There are methods to file a claim without your social.
I too have been a victim of Identity theft. I NEVER give out my social..even to doctor offices. I often get into arguments over it, but in the end always win.
Her, and your SSN is going to get out in the world. There is really very little you can do to stop it and I can guarantee that your daughter will give it out on a regular basis instead of going through the hassle of having places like this set up another number.
The better way to handle ID theft is to trust places, like this. And to keep an eye on your credit reports.
You're going to declare her college expenses on your taxes, so yes you do.