Microblog Memes

Since nobody has mentioned the actual reason for this phenomenon yet, the difference here is usually one of known vs. unknown gender/referent. (At least for practically all older speakers of English. Some younger speakers do seem to be able to use "they" grammatically to refer to known people. Changes in progress, woo!)

Your example is a perfect one: in a question like "whose umbrella is this?" we have no idea what gender the owner is, and so "they" is grammatical for the vast majority of English speakers.

Once the gender/referent is known, however, for many/most speakers of English (myself included), "they" becomes ungrammatical and the speaker must switch to "he" or "she":

"Whose umbrella is this? Did they already leave?"

"That's John's."

*"Oh, they need to come get it then." (The asterisk here is the common linguistic notation for ungrammaticality. This also assumes that both speakers are familiar with who John is. You can still get grammatical "they" after responses that refer to unknown people, especially with common gender-ambiguous names like Pat.)

So, for anyone wondering why many speakers, probably including themselves (if they're honest enough to admit it), seem to find known-gender singular "they" to be awkward/ungrammatical when supposedly "it's been grammatical for a thousand years", that's why!