Nah.
If you are going to vacation in San Francisco, you are not going to go to Oakland. Unless you deliberately do that.
I have not once visited San Francisco, looked around, and thought - d@mn it I am in Oakland!
So what if people commute? People commute from one place to another all the time. The question here, yet again, is about San Fransico, not Oakland.
I'll make it simple. Here is a map. See that lump on the left that the OP is asking about? That is San Francisco. See that water divide that you need to use the 80 bridge? Once you cross over that you are in Oakland.
San Francisco is not Oakland.