Is this because the kernel assigns that port to that specific process, so that all traffic at that port is associated with only that process?
Yes, that's what ports do. They split your IP connection into 65,536 separate communication lines, that's the main thing, but that is specifically 65,536 1-on-1 lines, not party lines. When a process on your PC reserves port 80, that's it. It's taken. Short of hacking the kernel itself, it cannot be reassigned or stolen until the bound process frees it.
The SO answer you found it interesting, I was not aware that the Linux kernel had a feature that allowed two or more processes to willingly share a single port. But the answer explains that this is an opt-in parameter that the first binding process has to explicitly allow. And even then, traffic is not duplicated to all listening processes. It sounds like it's more of a "first come first serve" to whichever of the processes are free to read the incoming message at the time it arrives, making it more of a load balancing feature that isn't a useful vector for eavesdropping.