Electrical – How does RFID reader reads a passive RFID tag

No, they're not globally unique, they're unique per manufacturer. For one thing, not all manufacturers use the same code length. If I want to start producing RFID tags tomorrow I can do whatever I please to do, like producing two RFIDs for every code, or even produce a whole batch with the same code. If you want to have them unique, you'll have to stick to one manufacturer.
That said, the codes are pretty long, so chances are slim that you'll have two colliding codes.

In the (near?) future RFIDs are going to replace bar codes in logistics. Those will have a combination of a non-unique product code, like EPC (European Product Code) and a unique ID.

I can only relate my experiences: -

If you want to detect a normally-not-powered passive type tag at extreme distances you have to power to that tag from a significantly bigger magnetic field. Making your magnetic field stronger is the only way I can know (and can recommend). Making your tag more efficient in recovering a fraction of this power is also part of the deal. Making the energy needed by the tag smaller is also part of the deal.

Once the "passive" tag is receiving sufficient energy from that magnetic field, it can transmit an RF signal to announce its presence - because it is only very weakly powered it may not be able to transmit more than a few hundred microwatts. This transmission should not have to do-battle with the prevailing magnetic field that powers it - it should be on a carrier frequency that is unconnected with the power magnetic field for this to work most effectively. This will require that the stationary object that generates the power magnetic field is capable of receiving this RF signal.

So now you have two transmissions - the transmission that powers the tag and the transmission from the tag containing ID data - neither are at the same frequency if you want maximum distance.

At about 4 inches (maybe 5 inches if I pushed it), a system I developed could detect the presence of a normally unpowered device. However, I needed to transmit about 1 watt across the gap because the device was doing other things that needed the power - it was rotating on a shaft and wires wouldn't work. The FM transmitter it used was at 80MHz and transmitted at about 1mW. The receiver could detect this at about 1m but it wasn't particularly designed to detect it more than 4 inches. The magnetic field it generated was quite large and the coil it used was wound from Litz wire - I reckon it was about 3 uH and had about 400 volts peak to peak across it at 600kHz (work out the current for yourself!!). Operating the magnetic field at 13MHz could be better but it starts to become a trade-off because, in your situation you want the "detection area" to be large - this means a large diameter coil and you want maximum current through it to produce the bigger and more far-reaching field you are fighting against the inductance of the coil. You need current in that coil to produce a magnetic field and the more the better.

To get that current, I used 250 strand Litz wire and parallel tuning to make the circulating current in the coil much much bigger than the drive current from the generator. This makes it easier to design the generator of course.

In short, if you want to power the tag at distance, think big coil and think litz wire and think parallel tuning for maximum efficency. The power receive coil was also very low loss and highly tuned to get as much voltage as possible when set at the maximum distance. This is what you should focus on in my opinion.