No they aren’t related but they share a common story.
“Tapatio” (m) or “Tapatia” (f) is a colloquial term to describe someone from the highlands of Mexico or from Guadalajara. Both are located in the state of Jalisco.
For the tequila, Jalisco is the premium growing region of blue weber agave and home to the most prestigious tequila distilleries, including Tapatío.
For the hot sauce, the company’s founders emigrated to America from Guadalajara, the capital of the state of Jalisco.