Purpose of the magnetics center tap in Ethernet
There are two sides on the transformer.

On the PHY side, the center tap is often connected to the supply voltage, for example, with 100BASE-T MLT-3 encoding, the PHY has to pull either side of the transformer to generate a positive or negative pulse.

On the "wire" side, the center tap can connected to ground (either directly or through a capacitor) to protect against EM perturbartions. It is also used in Power Over Ethernet setups to apply a power supply. It is better to use the center tap instead of one side, to balance the current accross the wires and inside the transformer (avoiding saturation and signal perturbations by having opposite currents through windings)