I'm an engineer. Typically I interviewed with various disciplines and bosses. Lunch was usually a softball interview to make a candidate more comfortable. The last interview I had, I was considered a high value interviewee (not sure if I deserved that) but it meant I was asked to give a presentation during lunch. It can be pretty brutal honestly. Some places are worse than others. I heard that Google and Agilent would sometimes ask candidates to solve equations. There's no way I could perform like that in front of people.

To bring it back to the topic, I do think it would be a bad look to bring in a Starbucks to the interview.