I have made some DC resistance measurements under no load. Hot was after roughly 1 minute of operation (under no load).
Terminal Cold Hot
230-0 169 288
115-0 115 261
0-22 1st 20.7 35.7
0-22 2nd 22.1 40.5
Primary AC current : 410mA
I have not found any shorts between primary and secondary (measured the DC resistance between every possible combination of wires). As you can see from the picture, every wire is in a different color and clearly specified, so I feel pretty confident that my wiring is not an issue. You can also notice that the label is damaged due to heat...

At that point I see these possibilities :
a) This is normal behaviour : it is just a little bit crap
b) This particular one has some kind of failure
c) This item is mislabelled and is not supposed to do what the label says it would
d) There is something really dumb that I did, and it is so obvious that I cannot see it
I hope to get an answer from the manufacturer for some documentation but I don't think it will be before next year. If anyone has an idea or something I could test I am still curious to know what is going on, but with only a multimeter at hand I don't think I can do much more.