The word "brand" originally meant "torch," and still does occasionally. To "brand" someone or something was to burn an identifying mark into the skin (or other surface, such as on a wooden crate or barrel). On the open range of the American west in the period from about 1870 to 1890, cattle belonging to different owners grazed together. To identify each animal, a geometric pattern was burned into the skin (usually when it was still a calf) with a "branding iron." Each cattle owner had his own "brand," and his home ranch was often known by a description of the pattern: the brand for the "Rocking R" ranch would be the letter R with a curve connecting the vertical and diagonal parts of the letter; the "Bar X" ranch would have a pattern that looked like an X with a horizontal line connecting the tops of the two diagonals. The pattern was made by a blacksmith from bar iron and fixed to a long handle. These "branding irons" could be heated in a wood fire and the heated surface pressed briefly into the skin of each new calf during "branding season."
"Brand" now usually refers to a special name for a company or product, and to the popular images associated with that name. We distinguish between "brand name" products that are widely known and advertised, and "off brand" or "unbranded" products that might be identical, but don't have a widely advertised name.
The writer used "side-of-the-steer brand" to indicate that he was talking literally about the earlier use of the term.