I think classic bags don't depend on brands. Some shapes are classic and these can easily be found within most big brands. What makes them classic is their longevity, high quality craftmanship (to ensure longevity) and materials as well as colours and embellishment that aren't trendy and keep their lustre.
Famous bags that fill all these requirements:
Birkin, bolide, drag and kelly from Hermès
speedy, alma, lockit by LV
but there are tons of others.