Why is the aqueous solution of KMnO4 slightly basic?
The strength of permanganic acid that you quote, combined with that of potassium hydroxide as a base, would guarantee that pure potassium permanganate is neutral in aqueous solution. But commercially prepared potassium permanganate is made in the presence of alkali, the use of potassium instead of sodium arising from the fact that the reaction scheme does not work with the latter.
$\ce{2 MnO2 + 4 KOH + O2 → 2 K2MnO4 + 2 H2O}$ (in molten KOH solvent; reactants are fused together)
$\ce{2 K2MnO4 + 2 H2O → 2 KMnO4 + 2 KOH + H2}$ (electrolysis in alkaline media; a direct chemical reaction would convert some material back to $\ce{MnO2}$)
Samples are of course purified but must be assumed to contain some leftover alkali from this preparation.