A transformer
covers many items, but with electrics we normally consider wire wound units which isolate, change voltage, convert current into voltage, and also those which don't isolate all as transformers. Where there are extras then there is not real point where we can say it is no longer a transformer, be it a case, a rectifier, or a complete host of electronic components. Wire wound items also include chokes and ballasts and there is no real difference between a tapped choke or a auto transformer except for where they are used. Since the one in question is transparent you can see the transformer inside. Maybe a better term would be power supply?
However that does not help, what does help is knowing the smallest lamp it can power is 20W. So either the unit needs to power a group of LED lamps to get the 20W load, or the unit needs replacing. Today you can get pulse width modulated power supplies which go form zero to 60W so you could replace it with a unit able to work either a LED or quartz lamp.
In real terms there are three ways forward.
1) New power supply, be it toroidal core, new type PWM which will go to zero current, or a DC supply.
2) Re-wire so many LED's fed from one "Electronic transformer".
3) Move from extra low voltage (12v) to low voltage (230v) lamps.
The reason for 12 volt or separated extra low voltage (SELV) was two fold.
1) To extend the life of the quartz lamp.
2) Extra safety in special locations like bathrooms.
If the latter then you are stuck with extra low voltage, but with the former using LED all the advantages of using ELV have gone, in fact I have found the 230 volt LED lamps last longer than the 12 volt versions.
There are special DC versions of the G5.3 lamp, mainly for use with caravans and the like, they do have a really good lumen per watt, normally 100 lumen per watt, and a voltage range of 10 ~ 36 volt. However they are expensive compared with the standard domestic model. The standard domestic model is between 60 and 80 lumen per watt, until reading the advert on this post I had assumed AC or DC, but that bulb clearly says 50Hz so is AC only. With AC they can use a capacitor to limit current, with DC that would not work, but with a capacitor the frequency would matter, so a switch mode unit working in the kilohertz range would be no good.
I think it unlikely that the lamp would use a capacitor, and think unlikely it really has to be 50 Hz, but it states 50 Hz on the bulb.
After the premature failure of my G5.3 spot lights after changing to LED, when no other LED lamp in the house has every failed, I would only use G5.3 LED spot lights if no other option. And I would try to use a smooth DC supply and DC lamps.