What more specifically is meant by giving the product a "personality"? A touch of distinction, perhaps even allowing it to be opinionated. But what does that mean on a more practical/concrete level?

Among other things, the personality bundles up the intangible aspects of the product in a way that helps potential customers figure out whether they love it or hate it.

What are some of those intangible aspects? Any examples?

How does the product make you feel when it works well? And how about when it breaks or wastes your time?

So are we talking about an Octopussy like Github and a Fail Whale like Twitter? Cutesy personality stuff like that?

May work for some... and certainly throwing down a comical quirkiness to your failings helps to ease the pain of being down (Fail Whale, Blippy's rainbow 404 page - not site failing but still cute) but one must be careful with how they go about giving a personality. Too cute and missing the target market and I'd wager you'd be dismissed as amateur.

If what you mean by personality is "theme" then, yes. Every site, every business, every aspect needs a consistent theme. It doesn't matter if you're a SAAS or a coffeeshop, a manufacturer or a lawyer. Every business needs a theme that demonstrates their "why".

That still doesn't answer the question posed, though.