Does getting any job (no matter how demeaning) builds up one's character?
I suppose much depends on what one means by "building character"; if it's increasing self-esteem, then the answer is probably no, as you've already pointed out.  But I don't think that's what most people mean by building character.  So what is it?  Certainly you'll learn something from having a demeaning job, because you'll learn something from just about anything; also, it is likely to lower your expectations, so you might be more excited about any given other job and value the money you earn more (see for example this article).  So it's probably possible to adjust the definition of "building character" until it's mostly true.  But after twisting the definition too far, it's unclear that "building character" is a desirable thing to do, so it doesn't offer much in the way of advice for life.

Incidentally, it has been argued that most work in the US is actually fulfilling.