I know the meaning of give up (abandon, surrender)  and break up (e.g.
break in pieces).
But I would like to know why "up" added to those phrasal verbs mean
apparently unrelated things. "Up" by itself doesn´t have anything to
do with these meanings.
What is the origin and metaphorical meaning of these words? 
WHY do they mean what they mean?

Just a free comment.  As a speaker of German, I immediately notice
that "give up", "break up", "take up", "write up", ... have direct
equivalents in German that mean pretty much the same:  "aufgeben",
"aufbrechen", "aufnehmen", "aufschreiben".

Etymologically, "auf" and "up" are related, and "phrasal verbs" are
much more common in German, called in English, "verbs with separable
prefixes", because the "auf" is a prefix when the infinitive is
written/spoken, but in other forms it drops to the end of the phrase.
(Ich will das Projekt aufgeben.  I gebe das Project auf.)

So, I expect that phrasal verbs reflect the Germanic roots of English,
the use of a preposition without an object in conjunction with a verb.

Oh, English uses "up" and other prepositions as a prefix to verbs: 
uptake, input, uprear, upkeep, upfold (yeah, I'm looking at the
dictionary).

Now I just hope our language expert, Scriptor, doesn't come and shoot
me all up  - or down ??

This might be of interest to you:

http://www.chass.utoronto.ca/~cpercy/courses/6361lamont.html