Indeed i personally know the guy who tried to open a bank account in Cyprus on his US passport and after many attempts was denied; he ended up opening an account on his Russian passport easily.
Which means he lied on application, because every European bank asks if you are a citizen of US.
IANAL but AFAIK no matter which other passports do you have if you are an US person (a citizen or a tax resident) you must notify you bank in any country to comply with the US laws (but having another passport makes it possible to hide that one is an US person).