Ruaḥ means “wind” or “spirit”—not “breath”.
Your friend may be confusing neshamah— “soul”—with neshimah, which is “breath”.
To your question as to why the two latter words are of the same root, the answer is because of the belief that the human soul or spirit derives from the breath of life that God blew into the human form when he made Adam (Gen. 2:7):
וַיִּיצֶר יְהוָה אֱלֹהִים אֶת-הָאָדָם, עָפָר מִן-הָאֲדָמָה, וַיִּפַּח בְּאַפָּיו, נִשְׁמַת חַיִּים
And the Lord God formed man of the dust of the ground, and breathed into his nostrils the breath of life; and man became a living soul.
(Originally written in reply to a question at Quora.com)