So are NFC tags really UNIQUE from each other, at least in their SERIAL NUMBER? And can we rely on the fact that no 2 NFC tags can have the same serial number? I'm highly skeptical about this as there are (and will be more) NFC tags out there and I don't think anyone is controlling the serials...
The reason I'm asking is that I'm developing a key based system using NFC tags. I don't need to write to the tags, I basically just need their serial numbers. But I need them to be truly unique.
Also, I would like to know if the serial numbers can be cloned (I know the content of tags can always be read and cloned, but can one program another tag to have exactly the same ID/serial?)
That depends on the tag product and what you consider truely unique. E.g.:
Yes, definitely. If you look for instance at MIFARE Classic, you can easily get cards with freely customizable serial number from a Chinese vendor of your choice. There is also devices like Proxmark with which you can emulate a card with a freely customizable serial number. And, moreover, it is relatively easy and cheap to build a device that emulates the protocol and consequently emulates a serial number of your choice. (Also see this question on emulating arbitrary serial numbers using Android NFC smartphones.)
As an attacker, the interesting part now is how you find a legit serial number for a specific system (i.e. a serial number that successfully authenticates to the system):
No, definitely not. Still there exist lots of such systems. But more and more of these systems introduce additional security mechanisms.