Shock waves develop inside the pitot tube under certain conditions, and that causes a spike (or sometimes a drop) in pressure which is interpreted by the instrument as a corresponding rise or drop in airspeed.
Edit: as for the 262 claim itself it was pretty handily debunked in a post-war NACA technical note. If you dig around on the NTRS you can find it.
https://en.wikipedia.org/wiki/Hans_Guido_Mutke
In other words, it's the same problem that the piston-engined aircraft of the time had: Anomalous readings at high speed.
For an airplane to reach Mach 1, this mainly requires thin wings and enough power/dive. To control it, you need a flying tail, and swept wings and "waisting" the fuselage reduces the amount of power needed.
So almost all of the planes in WW2 were "too thick" and incapable of supersonic flight, though could have exceeded Mach 1 with modified wings, tail and in a dive.
The US P-80 (1945) and F-86 (1949) jets had underpowered engines, in the sense that all early jet engines had limited power.
However the F-86 had swept wings, and later versions of the F-86 had a flying tail, so if you pointed it at the ground, you could get a sonic boom without losing control.
Source: I study aerodynamics.
Aerospace engineer and amateur WWII historian here: I'm not aware of any credible claims from that era. The only WWII-era claims to have broken the sound barrier (by piloted craft) I've seen can be attributed to poor pitot tube design which gives erroneously high indicated airspeed readings when the ram pressure gets high enough.
A reconnaissance Spitfire I believe is what got closest, at something like .9 of Mach measured by a fancy pitot system at the RAF test establishment.