The TIN is the Taxpayer Identification Number, also called an AFM in Greece. So you as the owner shook have that. I don't know why the renter should want or need it.
I don't know the laws in Greece, but in most states within the USA a renter whose lease is more than, say 28 or so days gains important 'tenant's rights.' These can be very onerous on the landlord. I don't know if long-term tenants in Greece gain substantial rights but I bet they do.
You would really want the guest to sign a written lease. A written and signed lease is really a MUST for you. You must state in your rules on your Airbnb listing that a separate signed written lease is a requirement for any rental exceeding x days. I would suggest that you also state in your rules on Airbnb that you would require the satisfactory passing of a background check and a security deposit of $X or y number of months. It's possible -- I don't know -- that a specific tax or 'stamp' tax applies to the lease, and the question is who pays it.
You really need to talk to a lawyer about ALL the issues involved before you accept any arrangement that would give the guest the rights of a tenant. Obviously you will incur expenses from the lawyer and if the lease does not go through, then you will not have income to immediately offset the expense (but you'll be prepared for next time).
One possibility, if the guest is open to it, is that you sign a one month lease, paid through Airbnb. Then after one month each you and the guest can consider whether they wish to continue. You could then do that directly with the guest and by that time you will have a greater comfort level with the guest, and the guest with you and the property. That addresses the guest's desire to pay month by month.
PLEASE get that lawyer. If You have to evict a tenant it could take a long while, during which you are not paid.
