when Al2O3 melts which bonds are broken?
Right, the text you are mentioning is not clear enough. It is not correct  to say that the molecule $\ce{Al_2O_3}$ exists and is separated from the next $\ce{Al_2O_3}$. Such a molecule does not exist. A sample of $\ce{Al_2O_3}$ is made of a superposition of $\ce{Al^{3+}}$ ions and $\ce{O^{2-}}$ ions. It is not made of a superposition of molecules $\ce{Al_2O_3}$. Now the fact that MgO melts at a higher temperature than $\ce{Al_2O_3}$ is due to the stronger attraction between $\ce{Mg^{2+}}$ and $\ce{O^{2-}}$ ions. For some reason, the attraction between $\ce{Al^{3+}}$ ions and $\ce{O^{2-}}$ is weaker.