The word is just under plus arm, and you can think of two pictures sharing that one phrase. One is a place: the soft hollow between your arm and your side, the spot a goes into and a is made for. The other is a path: a hand that stays below the shoulder, swinging forward from underneath and letting the ball go while it is still low. The first is about where something sits; the second is about how high a hand travels. They are held together by the words under the arm, and that is a way of remembering them, not a claim that one grew out of the other.
From the word’s history
Wiktionary records underarm simply as under- plus arm. It adds that the armpit meaning is not old: it appears in the 1920s, and is believed to have arisen as a euphemism in advertising. That fits where the body-part senses still live today, on labels and in health notices rather than in ordinary speech.
Wiktionary 記載 underarm 就是 under- 加 arm。它還提到，指腋下的那個意思其實不老：大約 1920 年代才出現，一般認為是廣告裡的委婉說法。這剛好對得上今天的情況——講身體的那幾個用法主要活在包裝、標示和衛教文字裡，日常講話的人還是說 armpit。
Wiktionary via kaikki, source_ref kaikki:3f6f69a190036ae4:1144656625: From under- + arm. The armpit sense is not old, historically speaking (1920s), and it is believed to have arisen as a euphemism in advertising.
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