Here’s a related ongoing thread discussing some of the same things from a different angle: Typecast - #12 by Just-B
A generalist has an awfully broad target audience, which brings up the problem of it being difficult to develop promotional materials, like a website or portfolio, that reflects a style conducive to everything from designing corporate annual reports to web sites for tattoo parlors. Almost always, though, generalists aren’t quite that general and work within a more restricted range of clients.
Even so, as the thread I linked to points out, as our career progresses, inadvertently or not, we tend to specialize due to short-term choices that lead us down long-term paths that present opportunities that branch off still further into increased specialization. Being aware that this tends to happen, whether planned or not, suggests the importance of making career and target audience decisions based on where you want to end up rather than choosing a willy-nilly course based only on what seems appropriate at the moment.
As for specialization, though, it seems increasingly common for designers to deliberately narrow down to a niche field where they can present themselves to clients as the expert in that niche. I think there are pros and cons to doing this, but one good thing is that doing so makes it a lot easier to focus in on a well-defined target audience.