iCloud doesn't have anything to do with Chrome. iCloud will sync
Safari bookmarks. Safari also has a button that allows you to view a list of
Safari tabs open on your other devices, but it doesn't synchronize tabs between devices in the way you describe. There's also Handoff, which will add an icon to the dock when another Apple device that's signed into the same iCloud account is in close proximity to the Mac, allowing that document/web page to be opened on the Mac. However, it does not automatically open it - it has to be clicked.
Google has its own cloud, and Chrome uses Google's cloud for syncing, not Apple's. See
https://support.google.com/chrome/answer/165139?hl=en&ref_topic=7439637
Now, there are good reasons for your daughter to have her own Apple ID/iCloud account anyway, rather than use yours - everything from separating your contacts and calendars, Safari bookmarks, passwords in iCloud Keychain, iCloud Documents, iCloud Photo Library (among other things), plus a separate identity for iMessage and FaceTime, and her own 5 GB of free iCloud storage space.