Not citizenship, but tax law. The US is the only major country in the world which taxes its citizens when they live abroad, so EU fcompanies need to know if you're one, so they can report to the IRS and tax you properly, or handle their anti money-laundering responsibilities.
Yupp. It's more a case of firms not wanting the hassle, so they usually just deny service to US citizens. Apparently transferwise has a similar issue, I'm not aware as luckily I'm not a US citizen.
Bunq was happy to open an account for me (US citizen). They asked for my US SSN and Dutch BSN at the time of account opening.
Maybe it's changed recently - I'll look into it again.