Is a Frappe actually have coffee in it?

Not necessarily. The Word "frappe" is used to mean different things in different places. In New England, it means what normal people would call a milkshake (a "milkshake" in New England does not necessarily involve ice cream; it's literally milk "shaken" ... usually, really blended ... with flavorings).

In other places, a frappe is a frozen drink (not necessarily involving either milk or ice cream), which may or may not use coffee as a flavoring... think "slush" here.

Starbucks "Frappuccinos" may or may not have coffee. Most of them do;as far as I can tell, the ones that don't are the tea-based ones (green tea and chai) and those that have the Word "Creme" in the name. If you're not sure, you could always ask before you order whether the one you want contains coffee or not.