- How would you characterize the short section at the end of a piece of music (climax)?
- What do you think is its function?
A short final section is often called the 'Coda'. As you say, its function is to wrap everything up after the main musical argument is completed.
The musical term 'Coda' originated with classical Sonata Form where themes were stated, developed, re-stated - and then you could finish of with a Coda. As with all such things, its definition has become looser over the years.
Your use of the word "climax" is confusing, because the climax of a song is not usually a short part at the very end. So I will focus on your main question instead.
In popular music, a piece sometimes ends with what is called a tag. This part is usually only a few measures long and most often repeats a line that was heard previously in the song (e.g., the last line of the chorus, the first line of the first verse, etc.). Songs may repeat the tag indefinitely during a fade-out or extended outro (e.g., "Pleasant Valley Sunday" by the Monkees), or they may use the tag once and end abruptly (e.g., "Love's a Loaded Gun" by Alice Cooper). The tag may, instead, have something of a kicker, an idea that puts the rest of the song in a new light (e.g., "Foolin'" by Def Leppard). I suppose, in this sense, it could be considered something of a climax, though I wouldn't call it that.
In any case, the purpose of a tag is generally to leave the listener with whatever idea (musical and/or lyrical) that the artist want to remain foremost in their mind.