"Kiu" is actually used to refer to something from a closed set. That's why we have the rule of thumb that it refers to people. There are only so many people in the world. However, it can refer to other closed sets -- like the finite set of possible telephone numbers -- which of these is yours?
The answer to "kio" is generally a noun, so the answer to "kio estas via telefonnumero?" would be "it's a series of digits which can be used to reach me by phone. In practice most people would just roll with it, and opinions may differ among fluent speakers of Esperanto.
I think of kiu as which also, but that makes it work here in my mind. "Which number is the one for your phone?" is kind of how I hear it in my head. (Although the literal translation would be "Which is the number of your phone?" of course...)
I think that... kiu is used like an adjective, when the noun is stated. kio is used in place of the noun. You would say "kiu numero estas kvin" or "kio estas kvin" ("which number is five" or "what is five"). Here, it's a little tricky, since "kiu" is separated from "numero" in the sentence, but it is my thinking that because it is still explicitly stated, and because the "kiu" obviously refers to a "numero", we use "kiu" rather than "kio".
... but I am merely a komencanto trying to figure it out, so a more experienced esperantisto can certainly correct me...
- Kio means what.
- Kiu means which or who.
Since there are only so many possible phone numbers, it's possible to conceive of this as a "which" question. Of all the possible numbers, which one is your phone number?
I haven't actually looked into it in this case, but there's something similar in terms of personal names. It can be either kio estas via nomo? or kiu estas via nomo? - and in practice, both are about as common.
I didn't know that "telefonnumero" existed, but the placement of the possesive ("via") seemed strange to me, if the question isn't about the serial number of model number of the physical phone device.
Would "Kiu estas via numero de (la?) telefono?" or "Kiu estas via telefona numero?" also be correct and closer to the intended meaning than "Kiu estas la numero de via telefono?" (maybe on par with "Kiu estas via telefonnumero?")?
To me, "Kiu estas la numero de via telefono?" also implies that the asked person has exactly one phone (or only one which connected to the mainline or extension with the phone number in question), while in reality, many people have several devices with a shared phone number.
Mi respondi al via tria frazo, das-g.
It's exactly the same situation in English: we would ask 'What's your (phone) number?' of somebody, even though there's a reasonable chance that person could be contacted on more than one number (eg mobile, landline, work mobile, work landline). I don't think I've ever heard somebody say 'What are your phone numbers?'
"kiu" is pronounced pretty much as one would pronounce the (there probably imaginary) word "kiu" in Italian.
That might correspond with how a stand-alone word "chi" followed by a separate "u" would be pronounced Italian. It's not how a single word "chiu" would be pronounced in Italian, as there, the "i" would be silent, while in Esperanto, "kiu" is two syllables and the "i" even has the stress. It's also definitely not like the "ci" in "ciao". The consonant at the beginning of "ciao" might be similar to Esperanto's "ĉ" and "ĉiu" is a different table-word meaning "everyone", while "kiu" means "who" or "which".
Listen to sentences with "ĉiu" on that one's Duolingo dictionary page for comparison.
The "i" in "kiu" and "ĉiu" (and in "iu" and other table words ending in "u") is sometimes pronounced very short and sometimes even with a voice color making it sound more like the "j" in German "Jäger". That's OK as long as the stress is still on the first syllable and therefore on the "i". But it's also fine to stretch it out longer, so you could pronounce it somewhat like "key ooh" would be pronounced in English.