I don't agree with this statement. US does no bar anybody to have multiple citizenships. When you naturalize and become a US citizen, US doesn't require you to give up any other citizenships. It only asks to renounce alligiance to any other country, which is not the same. Frankly speaking, most other countries also require you to renounce aligiance when getting their citizenship, so this is pretty normal.
No matter how many other citizenships you may have, US will always perceive you as a US citizen.
Even people born in the US and citizens from birth, don't lose US citizenship if they naturalize in other countries later in their life.
One can renounce US citizenship after becoming a citizen of other countries, but it requires effort from the citizen. At that point, US citizen will likely have to pay "exit tax" and several fees. The common descriptions of renounciation process include people saying US officials asking multiple times whether citizen really wants to give up US citizenship, and giving several warnings that citizen will lose certain privileges once US citizenship is renounced.