You can ban something without being a government. However Starbucks actually said[1] "Starbucks to Eliminate Plastic Straws Globally by 2020". You could argue that Starbucks have banned the sale of plastic straws in their shops, but they are probably not banning the use of them, should a customer bring one in from outside.
[1] https://news.starbucks.com/press-releases/starbucks-to-elimi...