Why do the first and second ionizations of a diprotic acid produce the same hydronium ion concentrations in solution?
This is saying both the $\ce{H2CO3}$ and the $\ce{HCO3-}$ are both donating $\ce{H+}$ to the water to form $\ce{H3O+}$. What it means when it says "The $\ce{[H3O+]}$ in the two ionization equilibria are one in the same" is to say that both of these reactions are occurring simultaneously in the same system, and that there is only one measure of the concentration of $\ce{H3O+}$ for the system.

It is not saying that both species are donating $\ce{H+}$ to the same extent. As a matter of fact, the $\ce{H+}$ being donated by $\ce{H2CO3}$ will keep the the $\ce{HCO3- + H2O -&gt; CO3^{2-} +H3O+}$ shifted almost entirely to the left, meaning $\ce{HCO3-}$ will be donating an extremely small fraction of the $\ce{H+}$.