Who owns a brand?

Risking overly scrupulous semantics, I personally believe that the consumer "owns" the brand in that we engage with a brand by, usually, buying the product which we then consume - eat, drink, wear, drive, watch, laugh or cry at, etc. The brand is the sum total of our experience, both intelectually and emotionally, with both the product and the trade mark.
The company that manufactures and/or markets the brand should probably be called the "author" of the brand.
So, if you drink Coke, you own the brands Coca-Cola and Coke. The Coca-Cola company is the author of both and the owner of the trade marks.
Here's a video from AdAge that discusses this dichotomy (using "brand" to mean "author," very specifically in this case) and illustrates how the line between authorship and ownership is beginning to blur and that Vuiton handbag counterfeiters and people who sell second generation CDs at swap meets aren't the only pirates out there.
Sometimes they're the most loyal fans of the brand that cross the line.