People in the bay hold Berkeley in a much higher regard than anywhere else in the country. If you go to Austin people will compare Berkeley to UT Austin, but if you go to Stanford they will regard your school to a higher degree.
I think OP should go with the better PI, however suggestion Stanford and Berkeley have the same level of prestige seems wrong to me
prestige among who? As a physicist, I couldn't care less what my school's prestige was amongst those outside of the physics community. Maybe you want to go work for a hedge fund, where Berkeley is also respected. Maybe it's worse to McKinsey? I don't know. Know your audience.
I think OP should go with the better PI, however suggestion Stanford and Berkeley have the same level of prestige seems wrong to me