What is the explanation of using pressure units (hPa) to express height when dealing with wind speeds?
There are several reasons. From the theoretical point of view it is beneficial to use isobaric coordinates, due to the vanishing density in the equations of motion. While in cartesian coordinates the wind is determined by the pressure gradient in isobaric coordinates the wind is determined by the geopotential gradient. The geopotential is defined by $\Phi = gz$. In the hydrostatic relation $p = \rho g z$ we can relpace $gz = \Phi$ and therefore any term containing $p/\rho = \Phi$. The geostrophic wind relation for example differ as follows:
Pressure coordinates: $\vec{V_g} = \frac{1}{f} \vec{k} \times \nabla \Phi$
Cartesian coordinates: $\vec{V_g} = \frac{1}{\rho f} \vec{k} \times \nabla p$
Notice the density as an additional variable in the latter formula.
Another reason arises historically. The vertical structure of the atmosphere could only be measured using balloons or radiosondes until around the seventies when satellites became an additional source of information. Until today radiosondes provide the most detailed profile of a vertical slice in the atmosphere. Radiosondes can not measure height in meter or feet but they can measure pressure. Also GPS is not be the best alternative. While it is quite accurate regarding the horizontal position it lacks accuracy in the vertical compared to pressure measurements.
Edit:
I forgot one of the most important aspects: Looking at pressure surfaces also allows us to compare heights in a more meaningful way. What we experience as weather happens to 99% within the troposphere. However, height of the tropopause varies a lot from poles (~9km) to equator (~16km). Looking at a pressure level allows us to make sure the mass of air above that level is approximately the same everywhere and thus makes variables comparable. Note that comparing what happens at 12 km altitude in the tropics to 12 km altitude at the poles does not make sense at all, because at the poles we are already dealing with the stratosphere, while in the tropics we are still way below the tropopause. Staying on a pressure level ensures that we take these differences into account.