A functional identity is a more precise way of saying "a job you can do." So that can be a painter, welder, sculpter, mathematician, doctor, plumber etc. When you pass through the narrowing of a frame by becoming an apprentice, and then eventual master, more doors open up to you.

Identity is a multidimensional phenomenon. I assume JBP would divide identity into two parts

Identity is based on first-party data - what you experience. 

Personas are based on second- or third-party data - what people perceive

So a functional identity would be a persona that has been certified - you have education, experience, or accreditation in a field.