Thank @Delphine348 for you reply, much appreciated.
Yes, I have seen that link and I think it is in fact the source of all the problems. I assume that the IT team at AirBnb took that data and created the forms.
The problem is that the TIN descried in the document is for individual persons only. If you look at EVERY example in the document it is for a real person, the document goes as far as to show passports and driving licenses etc. There is no mention or example of a business.
I can see the mistake, the IT team just assumed it also applied to businesses.
However, the document does tells us the name of a TIN in Romania, it is "Codul de înregistrare fiscală", or CIF for short.
In Romania there are two types of CIF. For a individual person it is a 13 digit number, normally based on the person's CNP (codul numeric personal). Everyone has a CNP written on their birth certificate.
However the other format of the CIF is for businesses and it is based on the CUI (codului unic de înregistrare). Every company is given a CUI at its time of formation. It is the unique business number. It is used in Romania for everything related to business: invoices, taxes, dividends, etc.
Here is a Romania web page about the CIF, unfortunately there isn't an English page on Wikipedia about it: https://ro.wikipedia.org/wiki/Cod_de_identificare_fiscal%C4%83
There is also this letter written to answer a question about the CIF and CUI from 2020. It is also only in Romania: https://commons.wikimedia.org/wiki/File:Mfinante-cif-1.png
The important bit is this (translated): "the CUI, respectively the CIF, represents the identification attribute of the respective entity." In this case "entity" refers to sole traders, and limited companies. In other words the CIF (the "TIN" that AirBnb and DAC7 are looking for) isn't the 13 digit number for persons, but a shorter number based on the CUI.
Once you know about CUI and CIF, searching the web for those brings up thousands of pages as they are used daily by every business.
As I mentioned in my original post, I have exchanged several emails with my accountant about this and there is no 13 digit fiscal number of any kind for businesses in Romania.