Q:

Why are mutiple kissES exchanged (at least one on each cheek) when "faisant LA bise"?

If “la bise”and “le bec” are both singular nouns meaning “the kiss,” why do French and many Francophone people generally exchange multiple kisses on their cheeks when they “faire la bise / le bec” upon greeting each other? (It's my understanding that some regions kiss up to four times [two on each cheek], so I don't think the "la bise" in "faire la bise" could mean "one kiss on each cheek".) 
Could it be related to men's exchanging just the one singular “bise/bec” with fellow men, which would seem to be called for grammatically by the expression “faire la bise/le bec” (or do men skip even the singular “bise/bec” in favor of other forms of greeting each other)?

A:

In french, le and la don't necessarily behave as counting markers for 1. They can have a general meaning. 
Le chien aboie = the dog is barking OR Dogs bark.
Generally dogs bark, billions of dogs, but you refer to it as a general fact with a singular defined article - le, la.
Still, you could also, in less formal style, say Les chiens aboient, or Tous les chiens aboient (every dogs bark).
(While Un chien aboie will usually be A dog is barking, but in oral style you will more casually say Il y a un chien qui aboie, There is a dog barking.)
Actually many french expressions, including articles, can recover various meanings depending on use and context. 
As far as la bise is concerned, you can even say 
Je passe te faire une bise dans l'après-midi,
Which doesn't emphasize that you're going to give 1 or 2 or 4 bises, but just use the bises as a métonymie for "passing by and saying hi", with some kinder or more affectuous feel.
Faire la bise is a general way of saying hi with one or more kisses on cheeks
Faire une bise will be one iteration of this coutume, but it can mean one kiss as well as ten, or even not especially any, depending on the context. 
Just like Having a drink will not necessarily mean drinking one drink, but drinking together, or more generally meeting / chatting, would you even order nothing or have dinner.
For your technical details question :
Usually, in Paris you make 2 bises, more looks very boring and unusual - except if you just want to make as many as you can to this very person to express your affection. Still, if you're close people, 1 bise only can mean the same (but you don't have the pleasure of many cheek-kissings),or adding hug, or hugging instead of kissing.. (Hug is less common in France than in US or Germany, but when you hug, you don't make tap tap with your arms, you really : hug.)
Outside Paris, uses will be variably 4, even 3.. it can also depend on 'friends / family / person you might finally kiss on lips', and most of all : on the mood of the moment, context and personal inspiration.
In french and in France, one rule only always applies, which makes the charm and beauty of it as well as its difficulty :
It depends. 

A:

Se faire la bise : s'embrasser sur les joues.
Biser : forme dialectale de baiser : familièrement donner la bise à (quelqu'un), synonyme  de embrasser.

L'embrassement (synonymes : accolade, embrassade, enlacement, étreinte) est une action qui commence par le rapprochement de deux êtres, se suit par un acte d'affection, et se termine par une séparation physique.
Se faire la bise (sur les joues), se donner une accolade (les bras [ou au moins un bras] autour du cou, [du col]), s'enlacer (s'entourer en serrant les bras), étreindre (presser dans ses bras) sont des actions uniques qui se décomposent chacune en action symétriques à gauche et à droite.
Dans l'usage français, faire la bise veut dire on s'embrasse selon la coutume locale, et non on dépose un baiser une fois sur une joue.
La bise n'est pas une, mais multiple selon les régions où on la pratique.
Parfois on demande « Je te fais une bise ? », parce que on a envie de n'en faire qu'une seule, et on peut avoir comme réponse : « Non, chez nous c'est deux [ou trois, ou quatre] ! 

Références tirées du  petit Robert

