That's right. This is by design, not a bug.
Pixels per inch is a way to translate pixels into a physical size measurement where it's needed. On screen there is no need for it, because a pixel grid is already defined by the physical screen itself. The image pixels just align to the physical screen pixels. That's your resolution on screen.
Export and Save For Web are intended for screen, not print or anything else. So they both strip the resolution metadata from the file altogether. An exported file doesn't have a ppi number at all; not 72, not 96, not 300, not anything.
Most applications, however, will assign a default number if there isn't one. Photoshop assigns 72, Microsoft applications assign 96. The file might be printed, or font sizes (a physical measurement) might need to be defined. So they assign a number, but it doesn't really matter what the number is as long as there is one.