To address your edit, some other countries only allow single citizenship. So if they get US citizenship, they are required to renounce their previous citizenship by their former government.
When you become a naturalized citizen of the US you have to renounce your allegiance to other countries, but you don't have to give up your other citizenship.
Yes, I'm clearly a horrible monster for believing something I've heard several places. No other person alive could ever have committed such atrocities, and I need a lecture from someone who has certainly never done something like this.
I'm not sure you are wrong. I've seen a room full of people recite the oath of allegiance. I don't know if it's legally binding or not, but it absolutely looks like a renoucement to me.
I hereby declare, on oath, that I absolutely and entirely renounce and abjure all allegiance and fidelity to any foreign prince, potentate, state, or sovereignty, of whom or which I have heretofore been a subject or citizen; that I will support and defend the Constitution and laws of the United States of America against all enemies, foreign and domestic; that I will bear true faith and allegiance to the same; that I will bear arms on behalf of the United States when required by the law; that I will perform noncombatant service in the Armed Forces of the United States when required by the law; that I will perform work of national importance under civilian direction when required by the law; and that I take this obligation freely, without any mental reservation or purpose of evasion; so help me God.