Sie wurden guten Freunde und lebten glücklich miteinander zusammen.
I found that miteinander and zusammen have exactly the same meaning. Why are they both used here? What kind of grammar is behind this?
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Well to me it seems that the story is a tale and this tale has been manipulated/simplified to a version with very easy vocabulary this is one important point.
One can say the reading voice is reading in a strange manner, but maybe it's just that she isn't used to as for little children and beginners one should read and pronounce carefully this is a challenge even or maybe especially for a native speaker and she is definitely one!
I guess i would do it worse then her! And i grow up in a region of germany where the "best Hochdeutsch" is spoken!
And i would finally agree with @Chris: the focus on one hand is on "lebten miteinander" = "they lived together" or on "lebten zusammen" and on the other hand "they where happy with each other" that would be "zusammen glücklich" or "miteinander glücklich".
And yes there is a redundancy but one must consider the above aspect that the story or tale has been simplified for beginners and maybe even for children! In that cases you may find some kind of redundancy because the focus is to use easy vocabulary and let them understand and learn these easy words!
And finally a typo: "Sie wurden gute Freunde und lebten glücklich miteinander zusammen.":
gute not guten Freunde
This is a pleonasm (https://en.wikipedia.org/wiki/Pleonasm). Its semanticly correct (Typo already said) from Medi Saif:
"Sie wurden gute Freunde und lebten glücklich miteinander zusammen
It makes the aspect stronger, that they lived TOGETHER.