ASML bought out their US competition decades ago. Nikon and Canon used to be their main competition in lithography machines, but both of them fell behind even before EUV.
ASML's EUV tech was originally funded by the US (EUV LLC, joint industry/government collaboration from the late 90s), and that is why the US can still impose export restrictions on ASML to China.
Government/industry consortiums on both the US and Europe sides ended up joining efforts. EUV LLC on the US side, EUCLIDES (Extreme UV Concept Lithography Development System) on the EU side (in which ASML and Philips was involved). The EUV LLC part is why the US can still prevent export to China of ASML's EUV technology.
ASML didn't come out of the US, but rather parts of their EUV technology did and the US still holds some rights to export controls through that history.
