I have no real idea why both sign with the same sponsor
They both sign the same sponsor because of the rivalry between the two teams. Whether that is to discourage one set of fans boycotting or otherwise causing mischief to the other sponsors product/company I do not know. But I suspect it makes economic sense to the sponsor to sponsor both and I suspect that it avoids trouble.
The fans really will not buy from a company if they sponsor the opposition.
As much as I see the logic behind what you say I can't really believe that people would change their drinking habits because of a football team. They never used to have the same sponsors and the only trouble I ever recall was when Celtic were sponsored by a car dealership who just happened to have a dealership right next door to Ibrox and the green and white flags which the dealership tried to fly didn't go down too well on match days.