Why is solubility often listed as a chemical property? Isn't the action of dissolving just a physical change?
The classification of physical and chemical changes has done more harm to students than good. This topic was taught in the 19th century but it still lingered in modern general chemistry textbooks which serve to repel good students from taking up chemistry as their field. For example, if we call the charge of electron red, and the charge of proton blue, will it change any fundamental thing?

You can try to understand physical vs. chemical change as follows: If you are changing the physical form (liquid, solid, gas) of one or more component, then you can safely label it as a physical change provided you can bring the original form back.

You can melt NaCl crystals into a liquid, this is a physical change because we can cool it and bring back the original material. 

In the same way, solubility is just a property of matter. It is a very complex phenomenon. Whether you call it physical or chemical it does not matter in real life (most scientists will not worry about it and nobody will ever ask you these questions once you pass this course-so much so for the utility of this concept). For example, HCl is very soluble in water, but also reacts with water in such a way that it protonates water. Dry HCl gas is "neutral" but its solution in water is a powerful acid. There is no simple way to extract the gas back out of water. This solubility involved a chemical change.

On the other hand, salt NaCl when dissolved in water, it will not chemical react with water per se and you can recover the salt by evaporation. Should I call this a chemical change or a physical change? It does not matter.