Why did the altimeter in an F-16 jump suddenly as it passed through Mach 1?
The altimeter doesn't measure static air pressure, it measures actual air pressure. So, to get an accurate reading of the static air pressure, the static port has to be carefully placed so that the ram air pressure (caused by the forward motion of the plane ramming into the air) and the slipstream pressure (caused by air rushing past the static port opening) cancel each other out (or nearly so).
So, Destin is correct: The actual air pressure after the shockwave is lower, causing a jump in the displayed altitude.