Why can soda be used as a substitute for eggs when using a boxed cake mix?

There are lots of people who have a rather simplistic approach to nutrition and think that removing fat and calories makes you healthy. Then they go through recipes for things they want to eat, replace the sources of fat with something which doesn't have fat and doesn't make the result outright inedible, and declare their recipe a success. I think this is what happened here.

In a cake, eggs provide leavening, moisture, smoothness, own flavor, and enhancement of other flavors. Oil provides smoothness and enhancement of other flavors (and possibly its own flavor, if not netural). And while it is not water based, it keeps the moisture in the cake from evaporating, so it makes the cake less dry.

If you are a "simplistic nutritionist" without all this information, you can approximate some of the effects with soda. It will provide moisture, and it will also provide some leavening because it is fizzy. It will provide some flavor of its own too, but frankly, I find the rather chemical flavor of soda to be unpleasant. And it won't have any fat. In the eyes of the simplistic nutritionist, it has successfully replaced the oil and eggs while reducing fat and calories.

From the point of view of a baker, the cake will be a disaster, and won't even deserve the label cake. It will dry out quickly because it has no fat. It will have a bland flavor. Its texture will be terrible. They say "more chewy?" It will miss both the protein structure and the emulsifying agents provided by the eggs. It will be essentially an overwhelmingly sweet quickbread with no redeeming qualities. From a culinary point of view, it will be terrible.

Bottom line: under some assumptions, it is a good substitution. For me, these assumptions are so far from reality as to be useless. It is a terrible substitution.

I have developed an intolerance to eggs and sought some advice from a vegan Indian family I know. They have always used soda in their cakes. Since they've never had a cake made with eggs they admitted they have no way to compare the cakes but they said it is spongy and not dry. For them it isn't a matter of simple nutrition but a cultural and religious diet.

Cake recipes that heavily use syrup (especially inverted syrup/honey due to its non-crystallizing, moisture-keeping properties) and can work well without eggs are not uncommon (several types of Gingerbread/Ontbijkoek/Syrup Cake...)

Now soft drinks (not the diet type suggested in that recipe, oddly, unless it uses sugar alcohols) ARE (thin) syrups. Additionally, the carbon dioxide will help start gas bubbles for raising, while the acidity of still dissolved CO2 probably helps to activate any (baking, not soft drink) soda in the recipe .

i am NOT a good cook...but COULD do cakes with eggs, oil...etc then I switched to the cola in the brownie idea...wow !! flavor, moist, not dry...not as 'heavy, dense' but much lighter and much much tastier...so those who wish to debate the science...I suggest you make a cake with just 2 ingredients...a box mix and a soda...give it a FAIR trial...don't tell the family (they won't know) until after the last crumb is consumed !!!