BMI isn't a "real" thing, it's an arbitrary measure meant to capture some aspect of "overweightness". The exponent has traditionally been chosen as "2" because that roughly fit with data observed. In the Keys et al 1972 paper that established BMI, they did try other exponents 1 and 3, but found 2 to be the best correlate of body fat.
Cubed relationships between length and weight are only an approximation, and species vary quite a bit on how close they are to the "cube rule". For a real-world application, I've come across this in the context of sport fishing, where it is easy to measure a fish's length but sometimes more difficult (you need a scale; the fish needs to hold still) and possibly harmful to the fish to weigh them; you can estimate an approximate weight from a species-specific formula, however. Wikipedia has a page on this with some examples for different species.
For humans, we simply don't tend to follow a cubed relationship. A XX% change in height for humans is not associated with an equivalent XX% change in width or "depth", it's associated with something a bit less.
There have been some suggestions to use a different exponent than 2, though, because with the current formula, BMI tends to not track well with adiposity or health outcomes for the tallest or shortest individuals. A barrier to making this change is agreeing on which one to use and the inertia of a publication record on the old measure. Here are a few examples, though, where people have investigated whether a different exponent for height and/or weight would better index a healthy/unhealthy body composition (I'm sure there are many more; not all ultimately recommend against the current scaling):
Foster, D., Karloff, H., & Shirley, K. E. (2016). How well does the standard body mass index or variations with a different exponent predict human lifespan?. Obesity, 24(2), 469-475.
Garn, S. M., Leonard, W. R., & Hawthorne, V. M. (1986). Three limitations of the body mass index. The American journal of clinical nutrition, 44(6), 996-997.
Tjeertes, E., Hoeks, S., van Vugt, J. L. A., Stolker, R. J., & Hoofwijk, A. (2017). The new body mass index formula; not validated as a predictor of outcome in a large cohort study of patients undergoing general surgery. Clinical nutrition ESPEN, 22, 24-27.
Xu, Y., Yan, W., & Cheung, Y. B. (2015). Body shape indices and cardiometabolic risk in adolescents. Annals of Human Biology, 42(1), 70-75.