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?
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.