「顔」 itself has a metaphorical meaning just as described in that paragraph. One can have more than one face in phrases/sentences like these:

> 表の顔と裏の顔 (lit. "front face and back face". The face you show to the world, and your inner side.)

> 彼は別の顔を持っている (lit. "He has another face". He has a secret hobby, or he is famous in two different fields, or he is a spy sent from an enemy, ...)

> アマゾンはクラウド企業としての顔も有している。 (lit. "Amazon also has a face as a cloud company.")

I think I occasionally see expressions like "女は2つの顔を持つ (Women are two-faced)" in Japanese literature. However, having more than two faces is rare, and I can't think of the Japanese proverb that exactly matches "Every person has *three* faces".