A typical lavatory will often contain two paper products.
Toilet paper is intended for cleaning your nether regions. It is soft (usually!) and is not intended to retain any strength when wet. It is actually designed to turn to flushable mush on contact with water, and to essentially disintegrate the longer it stays in water. As such, the toilet paper on the plane is safe to put down the toilet.
Paper hand towels on the other hand are usually less soft. They have a high fibre content, and are designed so that they do not lose significant strength when wet. As a result, paper towels will easily block up a toilet, and must not be flushed. This is what the sign is referring to which your daughter spotted.
Toilet users will often also use other disposable cleaning products. Tissues, wet-wipes, sanitary towels and so on are all designed to retain strength when wet, and again will easily block up toilets. It's worth noting here that even so-called "flushable" wipes have been found on testing to retain significant strength and to be a very real problem for waste disposal systems. The sign also relates to these.
Chemical toilets in caravans and boats have smaller-diameter pipes, generally with a macerator to break up "solids", and regular toilet paper can block this very easily. Some people use regular toilet paper and provide a bin. Alternatively it is possible to buy special toilet paper which breaks down much more readily when wet. Coming back to your airplane toilet, if the airplane toilet needs this special toilet paper, you can assume that this is what the airline provides. If for some reason you have your own toilet paper though, you might want to be cautious and dispose of that in the bin.
Regarding your point about toilets in other countries which cannot take regular toilet paper, the mush of wet toilet paper can still clog up narrow pipes, especially with longer pipe runs which may not have enough gradient to keep things moving. In some countries with smaller-diameter or lower-quality sewer pipes, toilets therefore have a bin for toilet paper. (In Europe, Greece is the main place you would see this.) Whether modern sewage systems genuinely still have problems with toilet paper is unclear - it is entirely possible that the previous technical issues have set up a tradition.