Acidic KMnO4 oxidizes H2S to S. Why doesn't the oxidation proceed further to sulfur oxide, sulfur dioxide or sulfur trioxide?
Yes, the reaction can further proceed to sulfur dioxide and sulfur trioxide, but only appears for certain, per my research, to occur in alkaline conditions.
Here is a supporting opinion to this effect, which cites the corresponding reaction in alkaline conditions as:

$\ce{3 S(2-) + 8 KMnO4 → 3 SO4(2-) + 8 MnO2 }$

that implies the further conversion of $\ce{S}$ to $\ce{SO2}$ to $\ce{SO3}$, which in alkaline conditions is represented by sulfate formation.
Note, the cited source notes only in acidic conditions, to quote:

Neutral-Acid pH:
$\ce{3 H2S + 2 KMnO4 → 3S + 2 MnO2 }$

Wikipedia on KMnO4 supports this pH effect, to quote:

The Mn-containing products from redox reactions depend on the pH. Acidic solutions of permanganate are reduced to the faintly pink manganese(II) ion (Mn2+) and water. In neutral solution, permanganate is only reduced by three electrons to give manganese dioxide (MnO2), wherein manganese is in a +4 oxidation state.

Interestingly, the corresponding reaction replacing KMnO4 with either H2O2 or HOCl, can results in $\ce{SO2}$, ..., in acidic conditions.