In Under the Nanoscope: What Is (Not) a Brand? series, we select one definition of a brand from a well respected source and examine it under our nanoscope.
A brand is a symbol that leaves a mental picture of the brand's identity.
— Source: Leo Burnett (2011)
Argument #1: Brand Is Not a Symbol
A brand in here is just a proxy of a symbol. But symbol is already defined term. Symbol is not a brand no matter how many mental pictures it will leave you with.
Argument #2: Modern Way of Marking Cattle
The part "...a symbol that leaves a mental picture..." is referencing to the practice called as "branding", marking cattle with hot iron. In this phrasing you don't mark cattle with a hot iron but customer's brain. Forever. But we know for a fact it is not possible this way. No one can mark a brain forever. Though agencies wish it to be true.
Argument #3: Agency's Sales Pitch
The part "...a mental picture of the brand's identity." is nothing more than sell of agency's services. Basically the statement says a brand is a symbol which looks like a brand identity.
This is not a definition of a brand. You can't predict any results by the statement. You can't guarantee any results. You can't measure it. You can't replicate it. This does not prove existence of a brand. The statement is just an agency's sales pitch.