You will receive a billing receipt, after you pay, by email. You can print it yourself or view it online at any time.
The receipt includes the traveller's name, the dates of the stay, a reference number, a description and address of the place you stayed, the host's name and contact information, a breakdown of the charges, the total cost and the payment method.
I have claimed expenses back from several public sector organizations by using AirBnB receipts, but not the Greek government. I have not had any problems. But, I know that the Greek authorities can be meticulous to the point of exasperation in desiring very particular paperwork. If you haven't claimed from them before, I would advise you find out exactly what paperwork will satisfy them.
An invoice in English is a formal document requesting payment, issued before the payment is made. You will not receive an invoice, only a receipt proving you paid, after you pay.