I love shitting on Fullstack devs as much as the next guy. However, sometimes it really just does make sense for an (often internal) product maintained by a one-person team, and it doesn't have to mean that the organization doesn't value them. I've seen it happen.

However I would not recommend it as a career path because it's essentially impossible to tell what you're getting into when you get hired. Could be what I just described, could be that you inherit the full responsibility for a 20 year-old perl+php5+xhtml+angularJS mess.
I think it can only truly make sense if you work independently and get to build projects to your own quality standards, assuming you manage to find a "scope is small enough that specialization doesn't make sense" niche. This is very hard which is why in practice "full stack" tends to mean "master of none but good enough to get a product out the door cheaply".