Can the titrant and the unknown material be flipped? We know the typical experiment of determining the concentration of $\ce{Fe^2+}$, where we titrate $\ce{MnO4^-}$ which has dark purple color and watch the color of the bottle.

Can we also determine the $\ce{MnO4^-}$ concentration using $\ce{Fe^2+}$ as titrant and how?



You can see that the iron is inside the beaker and we titrate with  $\ce{MnO4^-}$. Can we do it backward, i.e. having $\ce{MnO4^-}$ inside the beaker with purple color and we titrate $\ce{Fe^2+}$.
The titration reaction does not know or care whether one reagent is in the buret and one is in the conical flask (Erlenmeyer flask, it is not a beaker in your figure). And certainly you can standardize/titrate potassium permanganate with ferrous ammonium sulfate solution to determine the concentration of permanganate ion. Ferrous ammonium sulfate hexahydrate is a primary standard in analytical chemistry.

If you are extraordinary careful, you can fill the iron (II) solution in the buret and have permanganate in the conical flask. The end-point will be slightly vague (deep purple to colorless). You really have to be extra-cautious. The question is why bother with this painful eye-stressing procedure? Always take the permanganate in the buret.

A dark solution can be used in the conical flask. For example, in iodometric titrations, we start with a deep brown solution, which fades, to nearly a colorless solution upon titration with thiosulfate. In order to confirm the end-point, you would put a drop of starch before your expected equivalence point. Again, the color change is from deep blue to colorless.