Are the CL Final tickets sold by the clubs (as opposed to those UEFA sells) nominative? Like, do they ask you at the counter for an ID card or a name?
Yes they are.
Why must I bring my ID card or passport with me to the Final?
You need to have your ID card or passport with you on the day of the Final so that your identity can be checked against the details provided with your application. Should a ticket holder not be in possession of a valid ID card or passport, they will not be allowed into the stadium.
And from the ticketing terms and conditions:
3.7. In the Application Form, for all Applicants and Guests: (i) all Applicants are required to provide: surname, first name, date of birth, country of birth and Passport or ID number; and (ii) Italian nationals only, additionally, shall also specify province of birth and city of birth.