Repetition plays an important role in "The Other." On one level, it emphasizes the fact that the person the speaker is addressing is never satisfied; the poem's repetitive language reflects this person's repetitive taking from the "woman" they so envy.
For example, listen to the epistrophe and diacope in lines 1-3:
She had too much so with a smile you took some.
Of everything she had you had
Absolutely nothing, so you took some.
Epistrophe (the repetition of "you took some") emphasizes this continual theft; this isn't something that happened just one time! Rather, they "took some" again and again. The diacope of "had," meanwhile, emphasizes the way that this person keeps comparing themselves to the other woman, measuring whatever they "had" against whatever she "had."
The polyptoton in line 8 works similarly, calling attention to the zero-sum relationship between these people. That is, the first person feels like only one of them can have luck, happiness, success, etc. at a time:
Because her great luck made you feel unlucky
The repetition of the root word "luck" highlights the perceived imbalance between these people.
The speaker also repeats language more broadly across the poem. Most obviously, the first and final stanzas feature similar or outright identical language in spots:
She had too much so with a smile you took some.
[...]
Absolutely nothing, so you took some.
At first, just a little.
[...] Which left her absolutely
Nothing. Even her life was
Trapped in the heap you took. She had nothing.
[...]
You had much too much.
Only you
Saw her smile, as she took some.
At first, just a little.
The repetition of "had too much," "absolutely nothing," "smile," and "at first, just a little" all reflect the reversal that happens between these two people: the woman who was stolen from in turn becomes the thief (of "happiness," "ambition," "life," etc.).