What does the "mass" of a box plot mean? 
By “mass” I mean the size of the two subretangles that make up the box in a standard box plot. I thought the mass quantified the observations below and above the median but if it did so every box plot would always be symmetric, because 50% of the observations would fall below and above the median. If so, then the mass would be the same below and above the median line.

A typical box plot has seven regions.
At the lowest end, (1) in my image, are the low "outliers". A standard way to determine these if by considering the points that are 1.5 interquartile ranges below the first quartile (25% point).
Next, (2) is the lowest point that is not an "outlier".
Next, (3) is the first quartile point. This is the point at which 25% of the data are accounted for.
Next, (4) is the median, the point at which half of the data are accounted for. This could be considered the second quartile, though I do not hear that term used.
Next, (5) is the third quartile point. This is the point at which 75% of the data are accounted for.
Next, (6) is the highest point that is not an "outlier".
Finally, (7) is the region of the high "outliers", often the points that are 1.5 interquartile ranges above the third quartile.

set.seed(2021)
x <- rt(1000, 4.1)
boxplot(x)

While half of the observations are below the median and half above (sort of...there is not just one way to calculate median, and you could have an even number of points), that gives no sense of the scale. For instance, consider the set $(1,2,3,4,5,98, 99)$. Half of the points are above $4$ and half below, but the points above tend to be much further from $4$ than the points below.

