To brand is to develop a unique identity for a specific provider of products.
A brand is an identity which sufficiently identifies a provider of products within one or more markets.
One or more of a brand's traits, or a compound of those traits, must create a sufficiently distinctive identity within a specific market. Other traits support or further develop that identity.
Legal (brand) names and trademarks are identifiers, not identities. (Identifiers are sharply distinguished from identity in MOT.)