We figured out what causes it: This user had resized his character portrait in the UI settings. As soon as he set it back to the default size, the issue went away.
I suspect this bug appears when the UI scaling results in a fractional value. When the UI refreshes (such as when going through a door), the scaling gets applied and if the result is a fractional value the system bugs out a bit. The next time you zone, it adds the scaling factor on top of the previous one and continues to do so with each new zoning. The result is that the portrait's scale keeps getting thrown farther and farther off.
That's a guess, anyway.
Regardless, if you encounter someone with this issue just check to see if they have scaled the UI. That will probably prove to be the cause.