No. Rogaine is the first brand of Topical Minoxidil approved by the FDA for the treatment of male-pattern hair loss in 1988. The patent for the product has run out and generic brands such as Kirkland, Growell and Regro have been available, usually at a lower cost.
The active ingredient Minoxidil is the same across product brands. The main difference is that generic brands would vary in their inactive ingredients.
This can change the consistency and texture of the final product. For example, you may find different brands being of an oilier or drier consistency.