It's likely that Gene got the idea of each crew having their ship's individual symbol on their uniforms from the Navy practice of enlisted personnel wearing a rocker on the shoulder of their right sleeve that has the ship's name embroidered upon it.
Also, U.S. soldiers of all ranks wear the insignia of their division (and the army to which their division belongs) on their uniforms, for quick identification of a soldier's unit when first encountered.
It makes sense from the above perspectives and was an interesting bit of Trek lore that disappeared with TMP. I always thought it was a bit hokey that Starfleet would change its symbol for one incredibly-famous starship (considering that there were probably incredibly-famous starships once a generation). It doesn't bother me to see that go away.
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The United States Navy: Life, Liberty and the Pursuit of all Who Threaten It!
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