His/her ISP uses Dynamic Host Configuration Protocol (DHCP) ,which centrally manages and maintains and assigns a pool of IP addresses. Every time this person disconnects from the Server, and then reconnects, he is assigned a new IP. Large ISPs who use this system can have a large number of DHCP servers in their network, and any of those servers can assign IP addresses. I'm not going to explain the whole process, but the DHCP lease process is called Discover, Offer, Request, Acknowledge or DORA for short.
And yes, the IP address is traceable back to the ISP who assigned it.
Sigh. Will you marry me?