To be the face of something is to be the feature, embodiment or recognised representation of the thing. It can apply to a person (e.g. J.R.'s comment of Steve Jobs as the face of Apple), or even to a concept, as shown in the example given in the following definition:
the face of sth
what you can see of something or what shows:
Poor quality is the unacceptable face of increased productivity.
[Source: Cambridge Advanced Learner's Dictionary & Thesaurus]
Longman provides a similar definition:
the face of something
a) the nature or character of an organization, industry, system etc, and the way it appears to people
technology that has changed the face of society
Is this the new face of the Tory party?
the ugly/unacceptable/acceptable face of something (= the qualities of an organization, industry etc which people find unacceptable or acceptable)
the unacceptable face of capitalism
b) the general appearance of a particular place
- the changing face of the landscape
[Source: Longman Dictionary of Contemporary English Online]
Note that this usage is not quite an idiom, since "face" itself has the following as one of its defined meanings:
a (1) : outward appearance
put a good face on it
a (2) : the aspect of something that is perceptible or obvious upon superficial examination
the theory is absurd on its face — Kim Neely
[Source: Merriam-Webster Dictionary]
The origins of the word suggest that its application in the above usages may not be figurative but, rather, that the specific application to the human countenance is only one facet (pun intended) of the broader meaning of "face". From etymonline:
Words for "face" in Indo-European commonly are based on the notion of "appearance, look," and are mostly derivatives from verbs for "to see, look" (as with the Old English words, Greek prosopon, literally "toward-look," Lithuanian veidas, from root *weid- "to see," etc.). But in some cases, as here, the word for "face" means "form, shape." In French, the use of face for "front of the head" was given up 17c. and replaced by visage (older vis), from Latin visus "sight."
From late 14c. as "outward appearance (as contrasted to some other reality);" also from late 14c. as "forward part or front of anything;" also "surface (of the earth or sea), extent (of a city)."