No, that statement is incomplete. Displacement measure the MASS of a object if it is less dense than water and floats in water. A human body floats in water, ergo, it is less dense than water and the weight of the body is exactly matched by the weight of the water displaced. If the object is more dense than water, then it can only measure volume.
The question is about what is being done in the calculation of body fat. The procedure.
The goal of the procedure is to determine body density. To calculate body density you need to know the volume and the weight.
Now you can quibble about weight vs mass, but for medical purposes, weight is generally measured by a scale. I've been weighed in a numer of doctor's offices, and the technology varies, but none have immersed me in water.
But the only easy and cheap way to measure body volume is by total immersion. It makes no difference how massive or dense a body is. Once it is totally immersed you have its volume. But weight is a separate and independent measurement.
Try to look at what is being done. You want body fat. There is some empirically derived formula for estimating body fat from density, so you want density.
Density is a ratio of volume to mass or weight, so you need both volume and weight. You get weight with a scale. You get volume by immersion.
And yes, you could do a combined procedure whereby you measure floating displacement and total displacement.