If someone else has the same name as you and has also copyrighted their name then both certificates will have validity.
You, as owner, have the sole author-ity to use your name. You cannot use your certificate to make claims on another person with the same name as yours and vice versa.
If a claim is made for authorship of the same name, then you will need to default to use secondary documents to prove ownership. This happens all the time with or without a copyright certificate but we usually only pay attention to this when we are incorrectly charged for tolls, fines and the likes, and then we stand up to prove the fine does not belong to us but to another with the same details. This happens very frequently with credit rating agencies. This and more about all kinds of documents are discussed in the forum.