Just to add, the "arm" and "arm" in English have very different roots.
One is from Old English and the other from Old French. Although they sound the same, they are not related in any way which is meaningful.
It is like "bear arms" and "bare arms" and "bear arms" (as in "carry weapons, short sleeves, and the limbs of a bear)...all sound the same, but it is just coincidence as all the words have different origins.