I'm interested in examples of when name-brand is worse quality, but yes, name-brand isn't always objectively better, and is often produced in the same facility.

As usual, it depends, so don't knee-jerk to all one or the other, if it matters to you, compare the packaging (it'll say where it was produced, so you can guess when it's the same product).

Sure, I'm just not sure if it's more or less often than when it's equivalent. It's frequent enough that you should be careful if quality is what you're after.