As with so many other companies in the age of the Internet, Koala is merely a software platform allowing people who are "selling" to link up with those who are "buying".
Uber, lyft, redweek, airbnb, ebay, craigslist, etc., etc., etc, etc..
Just a computer software program that some coder coded in any one or more of multiple programming languages like Python, Java, etc.
Some companies do zero checking to confirm you have what you say you have. Don't be surprised if that timeshare rental advertised on craigslist turns out to be fraudulent as you stand there in the resort lobby with your husband or wife and three kids after arriving from the UK.
Uber and lyft want to see the driver and his/her car and make sure that the driver is validly licensed, that he/she passes a background check, and that the car is validly registered (and is a relatively late model vehicle). After that, they would prefer to never again be involved in any driving transaction aside from collecting its cut.
So Koala at least does some minimal checking that you have what you say you have. So I don't fault them there.
So all you have to do is provide them some written confirmation of what you have and I think you'll be fine. You own the timeshare. Do you have the deed to be able to scan and email them? Do you have your annual bill that sets out precisely what you own? If you didn't keep a copy of the foregoing, your ownership is a piece of real estate registered in the city or county in which your resort is located. Might you be able to access that real estate or tax assessors site and print out applicable page(s), including perhaps the municipality-stamped deed?
Their data entry clerk needs a reservation number to enter into his new listing page. They used my timeshare resort's reservation number as their "Ref". Apparently, you already have that. But, if not, I don't see why they couldn't use your annual bill's Bill No. or some other unique number.