哥哥 in this sentence means "I". It is an amusing way to say "I". It implies an emotion but I can't find a good way to explain it at the moment.
If it is a young girl, she can say "妹妹" instead. But, sometimes, she might also say "姐姐" to show her maturity in a fun way.
If it were me, I would add a "我" after it, for example, 姐姐我开的店 / 哥哥我开的店 /妹妹我开的店.... In that way, no one will get confused.