As said in the comments there is presumably nothing like "Apperoseife" or maybe it refers to the manufacturer, but lets play Sherlock Holmes and take a wild guess:
clue a) IndustrieArchäologie 2/2014's title was "Geschichte der Waschmittel"
clue b) there are 3 green blocks
clue c) sounds like apperoseife, maybe a typo or company
clue d) turkish product carton
Therefore i think we are talking about Aleppo-Seife / Aleppo soap as its originated from syria, was invented around 800 and it is green.
clue e) Defne translates to laurel (Lorbeer)
clue f) Zeytınyaglı translates to olive oil
clue g) Sabunu translates to soap
Olive oil & laurel are the two main ingrediants of Aleppo soap. Elementary, my dear Watson.
As said before this answer was a wild guess for fun. But given the clues I'm pretty sure the "apperoseife" is some kind of Aleppo soap
Looking at the new photo we can find the german packaging
Sonderproduktion - Natürlich
Lorbeerseife mit echtem
Olivenöl aus Hatay
but ... most likely we will never know why it was called Apperoseife :/