Is my ID recorded when I check in to a hotel? Does this happen even for payments under 5,000 lira?

Yes. Under Article 2 of the Identity Notification Law No. 1774, your identity details are transmitted to the nearest law-enforcement unit through the identity notification system. The rule about payments under 5,000 lira concerns whether your Turkish ID number appears on e-archive invoices — it has nothing to do with the accommodation facility recording your identity.