What causes these odd-looking currents in the Tasman Sea?
Great question. These features are mesoscale Eddies and they are akin to hurricanes in the atmosphere. They are the ocean’s high or low pressure systems and are the result of instabilities. The cyclonic eddy (counterclockwise direction in the northern hemisphere) is called a cold-core eddy and the anti-cyclonic eddy is called a warm-core eddy.



Unlike destructive hurricanes, ocean eddies are important physical processes for the transport of heat and particulate matter. They stir up the ocean drawing up nutrients from the lower depths and fertilizing waters as they propagate. They are important for the transport of marine organisms in the offshore direction and support phytoplankton and zooplankton production.

In the Tasman sea, this region is known as 'Eddy Avenue' which speaks to their ubiquity and persistence. In Everett (2012) they identified 30 thousand eddies in a period of $15$ years.


  ... at a density of 61 cyclonic and 63 anticyclonic eddies per $10,000$
  km$^2$. On any given day there are 16.8 cyclonic and 17.8 anticyclonic
  eddies within the Tasman Sea. Tasman Sea eddies are typically smaller
  than the global average ($82$ km radii compared with $86$ km), have a
  faster rotational speed ($23$ cm s$^{-1}$ compared with $15–16$ cm
  s$^{-1}$) and have a larger Sea Level Anomaly($10$ cm compared with
  $6–7$ cm) (Table 1).


Generation:


  “When the current separates from the coast, it gets complicated. It
  starts to wobble; it meanders. Rather than going in a relatively
  straight path like the Gulf Stream in the North Atlantic, it walks
  like a drunk man.” Not only does the EAC wobble, it U-turns. [...] The EAC is a bit  anomalous.” About two-thirds of the current retroflects back up into
  the eastern Pacific, breaking up the EAC further. “It really is a
  current of eddies.” Such instabilities saturate Eddy Avenue with
  eddies.


As far as the El-Niño connection, I can only speculate. But generally, El-Niño will give rise to positive SST anomalies. The figure below shows recent positive SST anomalies which are relatively strong near the coast. Because ocean temperatures are higher ocean eddies can increase in size and strength. 



References

Everett, J. D., M. E. Baird, P. R. Oke, and I. M. Suthers. 2012. An avenue of eddies: Quantifying the biophysical properties of mesoscale eddies in the Tasman Sea. Geophysical Research Letters 39, L16608, doi:10.1029/2012GL053091.