How can magnesium hydroxide be a strong base and be sparingly soluble at the same time?
I think we should not conflate &quot;solubility&quot; with &quot;dissociation&quot;. They are different phenomena. The former has to do with a molecule breaking away from the other solute molecules and forming interactions with the solvent. The latter is a process which occurs after dissolution, whereby the solute molecules break apart to form ions, separately solvated by the solvent molecules. For a simple covalent substance, the two phemomena are clearly different.
For an ionic solid, the two may seem to occur together since when a soluble ionic salt is placed in water, its ions are in fact pulled apart by water molecules. In this sense, it is dissolving and dissociating at the same time. Hence, by the Arrhenius definition of acids and bases, insoluble hydroxides such as $\ce {Mg(OH)_2}$ may not even be considered to be strong bases. Arrhenius theory would tell us that the strong base is the hydroxide ion, that is produced from the dissolultion and dissociation of the salt.
Actually, in particular for some metal oxides such as $\ce {MgO}$, the first reaction it undergoes upon contact with water is $\ce {MgO + H2O -&gt; Mg(OH)2}$. Hence, we can rightfully say $\ce {MgO}$ is not an Arrhenius base, but it is the precursor to the weak Arrhenius base $\ce {Mg(OH)2}$.
I would like to disagree with the part of the answer you have referenced in your post because as aforementioned, dissolution and dissociation are the same phenomena for ionic salts. It is thus not correct to say that &quot;what does dissolve is still $\ce {100}$% ionised into [ions]&quot; and hence, say that insoluble (or sparingly soluble) ionic salts are strong bases.
To generalise, Arrhenius theory would consider soluble ionic hydroxides such as $\ce {NaOH (s)}$ to be strong bases while it would consider sparingly soluble ionic hydroxides such as $\ce {Ca(OH)2}$ to be weak bases. In other words, the strength of these metal hydroxides as bases is directly linked to their solubility and this makes perfect sense since dissolution and dissociation for an ionic salt are the same process.