Specifically - Jake Gyllenhaal's beard.
The term face pet was first coined by "lemniscate", on September 29th 2010, when posting on the popular Jake Gyllenhaal fan blog called Wet Dark And Wild.
The word combination face pet was engendered in reference to:
1) the tendency men have to subconsciously stroke and touch their face and/or facial hair; often a form of auto-erotic touching when flirting to indicate they're interested
2) the health benefits discovered by researchers at the University of Warwick with regards to stroking or playing with one's pet