Like in some other countries, this RUT identifier is only for tax purposes, including VAT. There is little information in English on how the system works precisely, but this RUT number field on the bus reservation system is intended at Chilean tax payers. VAT/sales tax is collected on sales between businesses (compared to some countries charging it only to end customers) so I suppose this RUT is more intended to businesses, which are required to have a RUT.
In fact, I tried booking a ticket on the Tur bus website and there was no issue leaving the field empty, to go forward to payment.
Foreigners are not required to have one unless they have to buy "major assets" (like a house, or a car). For buying bus tickets, that would not be needed - unless you already pay taxes in Chile, which I doubt if you're asking. Lonely Planet mentions needing the RUT only if you want to buy a car.
As a funny use of this number, in the case of Portugal (where receipts also have a Tax Identification Number (TIN) field), the government decided to run a lottery for people that pay the sales tax and give their TIN, and people could win fancy cars. The goal is to push people to pay sales taxes and limit fraud.