There is a gesture which consists of pointing with both thumbs upward. The phrase "the thumbs up" is the name of this gesture. As such, it is not really divisible into "thumb" and "up". We can think of it as having a long name "the well-known two thumbs up gesture", "the thumbs up" being is its popular nickname. Moreover, that "thumbs up" is a category representing manifestations of that gesture, so that we have "a thumbs up" ("an instance of the two thumbs up gesture").

My assistant gave me the thumbs up, indicating success.

= My assistant gave me the well-known two thumbs up gesture, indicating success.

Several spectators gave me a thumbs up as I tore through the finish line tape in first place.

= Several spectators gave me an instance of the two thumbs up gesture.

The difference between "a" and "the" is basically the class/instance distinction. "The thumbs up" means we are taking the point of view that this gesture exists as a single abstraction only: there is only one thumbs up in the world, and if we see two people at two different time and places give us "the thumbs up", it is the same entity showing up in two places. "A thumbs up" refers to the manifestation, where the point of view is to regard each one as a separate instance.

I received probably more than fifty thumbs up from passers by outside city hall, when I protested the closure of Jenkins Memorial hospital.

= The viewpoint that these are instances that can be treated as separate and counted.

Fifty people gave me the thumbs up.

= The viewpoint that this is one abstract thing, like "encouragement" or "support". Fifty people gave me one thing.

Interestingly, we cannot regard, say, "smile" as a class in this way:

Fifty people gave me {*the | a} smile.

Ah, but no: indeed we can. But the nuance is that this is some special kind of smile.

A: Three girls gave me the smile today, man!

B: Dude, what are you talking about. What do you mean "the smile"?

A: You know, the smile! Like this. ("A" grimaces like a monkey).

B: Oh, the smile. You devil, you! Did you get their phone numbers?