I'm asking about if the IP address of my router (my "public" IP address) is actually unique, i.e. it is not shared with other routers/hosts over the Internet.
If it is the case, I imagine that there will be some NAT policies applied by my ISP, which translates my "public" IP address and the port of my connection in some other IP:port pair (just like happens in local NAT).
Many operators use a version of NAT which is called by several names: Carrier Grade NAT, NAT 44, Large Scale NAT.
Essentially the operators use private addressing between the network core and the edge devices (i.e. your router) and then translate the private addresses to public addresses at the core of their network before traffic goes to another network.
There is not a one to one mapping between the private addresses and the public addresses, and even port numbers can be reused over time.
So to identify a given private address and hence user from the operator logs, you need to look at the public IP address, the public port and the timestamp (as the same public IP and port may be mapped to a different private IP at a different time).
Things start to get even more complex when you try to make the network devices as optimised as possible - there are mechanisms to try to assign to same public IP and port to a private IP that it has used previously for efficiency. This can give the impression to an end device that it has a static IP mapping in the NAT, but often this is not actually the case.
Finally, all the above is independent of any locate NAT you may have in your home or business - i.e. you can and probably will have a NAT translation on you local router for your locals devices also.