We recently made a cobbler by adding oil, water, and powdered yellow cake mix (from a box) on top of sugary fruit filling and it came of the oven crispy and delicious with that distinctive yellow cake flavor. What gives yellow cake its flavor? Even without the egg, this flavor is apparent, so it can't be entirely egg related, especially comparing yellow and white cake flavors.
It turns out this may be something of an inadvertent trick question. I have yet to test this but I get the idea that store-bought white cake and yellow cake batter powders both have a more similar cake battery taste than I originally suspect. The reason is pretty simple: the combination and ratios of sugar, salt, baking powder, and flour are probably not very different between the two powder mixes (don't be fooled by the yellow food coloring). Later, when the egg or egg whites and butter vs. oil are added it will of course affect the final cake texture but probably not the cake batter flavor.
Another possibility is that yellow cake batter mixes might have artificial butter flavoring whereas white cakes simply use vegetable oil for a lighter flavor.