I tried to enroll my company to Apple Enterprise program, but get message "Cannot enroll in Apple Developer program"
Yes, you can't specify the DUNS number during enrollment. This is part of the 'due diligence' that Apple has to do themselves, in order to verify that you are really representing a company, instead of 'impersonating' one. They might also get in touch with the person you mentioned to have 'legal authority' for this company. 

Basically, with the Enterprise Development Program, you're setting up a legal contract between your company and Apple, and the rules for that are more strict than if you just sign up as an individual.