Step 1: Recognise the cell by its description - a giant multinucleate cell with a grape-like cluster of nuclei found in reactive (hyperplastic) lymph nodes.
Step 2: This specific morphology is tied to one childhood exanthem: measles. The Warthin-Finkeldey cell forms in lymphoid tissue during the early, prodromal phase of measles infection.
Step 3: The biology fits - measles virus is a paramyxovirus whose fusion protein drives cell-to-cell fusion, naturally generating multinucleated syncytia in both lymphoid and epithelial tissue.
Step 4: Pair each wrong option with its true marker: Negri bodies for rabies, typhoid (Aschoff-like) macrophage nodules for typhoid, and no characteristic giant cell for rubella. Only measles owns the Warthin-Finkeldey cell.
\[\boxed{\text{Measles}}\]