Can dissolving be a chemical change?
Your book is glossing over some details.

There are three things occurring when you dissolve non-metal oxides in water (e.g., $\ce{SO2}$, $\ce{CO2}$, $\ce{NO2}$, etc.):


The gas simply dissolves in the water. You are correct that this step is a physical change.  How much dissolves depends on the partial pressure above the water and the Henry's law constant.
Once dissolved in water, some of the non-metal oxide will react with water to form an acid (sulfuric/sulfurous acid, carbonic acid, nitric/nitrous acid, etc.) and this is the chemical change that your book is referring to. There will be an equilibrium constant that will dictate the relative concentration of dissolved gas to the corresponding acid that is formed.
The acid (usually weak) will dissociate a little bit.  Again, the exact amount of dissociation will depend on the equilibrium constant, $K_\mathrm{a}$.


Using your example, we would have:

\begin{align}
\ce{SO2(g) &amp;&lt;=&gt; SO2(aq)}\\
\ce{SO2(aq) + H2O(l) &amp;&lt;=&gt; H2SO3(aq)}\\
\ce{H2SO3(aq) &amp;&lt;=&gt; HSO3^- (aq) + H+(aq)}
\end{align}

Similar things happen when metal salts dissolve in water (although not $\ce{Na^+}$ or the other group I and II metals). The metal cations will react with water and form a weak acid.