An AirTag is NOT designed for tracking people.
I am assuming that you initially set up this AirTag with your iPhone and it is associated with your Apple ID ... and that your daughter has her own Apple ID ... correct?
If so, then the AirTag is working as designed. That is, once the AirTag is no longer within Bluetooth range (around 30-40') from your iPhone, it will start beeping on a certain interval. In addition, if another iPhone is nearby, that phone will be notified that there is an AirTag present. They then can cause it to make a sound so that they can locate it. Furthermore, if this AirTag is on your daughter, AND she has an iPhone or Android phone, she will be notified that there is an AirTag following her.
How can you prevent this? Don't use it for tracking! It's that simple. If knowing her location is important to you, AND she has an iPhone, you can locate her iPhone (thus her) with it, through the Find My app.