Ha un gattone fantastico, pieno di peli e molto soffice.

[Soggetto] + avere + qualcosa
The structure with 'AVERE' indicates possession, i.e. the subject owns this 'something'. For example I can say: 'Ho una macchina fantastica', in this case I own the car, that is the car is my property. This structure is also used with some feelings, and to indicate the age of a person.