I think because "handmade" has so many meanings for so many different people, and many of the famous brands also have very special meaning for so many, that it is fair to consider this another one of those questions that is so subjective, and so closely tied to the very personal area of belief, that if it is "handmade" to you, then that is what is important!
As a purely practical and largely unrelated to "brands" or anything else in the world of ready-to-wear, it is certainly possible to hire an artist to make a one-of-a-kind bag to your specifications - or theirs - whose construction will involve the actual hands of that artist constructing the bag, with or without machinery, but entirely without label or affiliation with a specific company.