I would imagine by now there are some treaties between most nations that exchange such information. There is also passporting system. You gave them any sort of passport (at least here in US) and they will look it up in the computer. They have to be looking up some sort of database, right?)
I could imagine someone holding so many citizenships would be very valuable asset to government agency that hire spies.
"The United Kingdom – United States of America Agreement (UKUSA, /juːkuːˈsɑː/ ew-koo-sah)[1][2] is a multilateral agreement for cooperation in signals intelligence among the United Kingdom, the United States, Canada, Australia, and New Zealand."
I think you would be required to disclose your other citizenships when obtaining the U.S. one.
They wouldn't necessarily know about others if you got the American one by birth, though if you were born abroad you can be sure that's going to be in the profile.
I could imagine someone holding so many citizenships would be very valuable asset to government agency that hire spies.