How to know which process was using a port after it enters TIME_WAIT state?

No, the system just doesn't keep track of this.

TCPView shows quite clearly that closed sockets which fall in the TIME_WAIT state are instantly re-assigned to process 0 (System).

Sorry, I misread your question; the above answer refers to Windows systems.
However, I think the same happens on Linux systems, too.

In my case, tracing socket opens greatly helps me to spot TIME_WAIT states (since typically means that my application is repeatedly creating sockets)

Based on how i can identify which process is making UDP traffic on linux? , I tried auditcl. "-S socket" does not work in my ubuntu, but the code below (optionally adding -F pid=pgrep -x yourApplication ) is quite good spotting socket creations

auditctl -a exit,always   -S socketcall -F uid=$USER -F a0=1 -F a3=1  -k OPEN_SOCKETS