It's interesting to think about why this was considered offensive. After all, we have games like Civ V which is chock full of specialized units based on different cultures, and nobody sees that as offensive. So, what's the difference?
What it comes down to is context. In Civ V, you're not just playing as a specific culture - every game begins with a brief history lesson educating you about that civilization and the leader you're playing as. As such, you're assured that the civ-specific units (which, to drive the comparison home, include specially designed warriors for the Iriquois who look and dress like Native Americans and have abilities based on thriving in forests and jungles) weren't just made up because they looked cool - they've been given a modicum of research and respect, and what you're seeing makes historical sense.
What we see here in Bravely Second is not that. We have a job system in which one of the jobs involves dressing up like a sexy Indian. Seriously, just read these jobs aloud to yourself: 'Knight,' 'Singer,' 'Healer,' 'Native American.' It fails to make sense in context, and what you're left with is "My job is literally to dress up like what I think an Indian looks like and do the things I think an Indian does." Imagine if they had a 'Mexican' class. Would the 'offended' reaction be any less surprising? Why?
Sure, this doesn't address the point of why the West finds this more offensive than the East. There really isn't a case to be made on that, though - one side cares more about the difference than the other, simple as that. I think however that, at the end of the day, what we're debating is context.