In the question, shouldn't it be tiu since it's talking about some specific umbrella and not an abstract thing?
From what I know of tio/tiu, I agree. It would make sense to say "Ĉu tio estas ombrelo?" Is that (thing) an umbrella? But "Ĉu tiu estas via ombrelo?" - Is that (one) your umbrella.
For me it helps, at least while learning, to mentally add "thing" or "one" to know which is which is "indefinite that" or "definite that" – if it makes grammatical sense to call it that.
"Ĉu tiu estas via propra ombrelo?" would be more like "Is that one your own umbrella", as though you had to choose one out of a set. "tio" is fine, and has nothing to do with specific versus abstract.
You have a point. "Is that [umbrella] your own umbrella" would indeed be "tiu" even if the first "umbrella" is only implied. Strictly speaking, for it to be tio, it has to be more vague... as in -- that thing in your hand, is it your own umbrella?
You can think that "tio" equals to "tiu afero". So the sentence equals to "Ĉu tiu afero estas via propra ombrelo?"
The use of "tiu" should have a noun together, with may be omitted if it can be get from context. So "Ĉu tiu estas via propra ombrelo?" equals to "Ĉu tiu ombrelo estas via propra ombrelo?"
Both sentence can be used with the same meaning, however, if the object your are talking about those not look like a umbrella, only "tio" will bring the idea.
So, using "tio" thi writer can refer to a ugly umbrella, a broken one, or maybe a tie-umbrella.
I think it depends a little on context. In this case, I would say it's mostly emphasis... suggesting that it might be someone else's umbrella.
Ahaaa so you admit it!
In Croatia we have a saying - Tko prizna, pola mu se prašta! In translation that would mean something like - Whoever admits (doing something wrong) half (of that he/she has done) is forgiven :)