What is the function of this choke-like transformer on the PCB side of this RJ45 jack+magnetics connector?
It's a Common Mode Choke integrated in the RJ45 connector.

If there is a voltage spike on an incoming line the common-mode choke will prevent it damaging the silicon by transferring the energy of the spike so that it appears on 10, 11 and 12 together i.e. there isn't a differential spike.