How come? I would value a Stanford degree and working at Google on a resume. I wouldn’t know what to think about The White House or Air Force (like many posting on their site) on a resume.
It's specifically the Presidential or White House [0] fellowships that are prestigious, although if you have any job in the White House that's just as well.
What value an employer would derive from seeing that on someone's resume is probably the same as with Google or Stanford: it shows they got in, with the known/assumed quality of the institution vouching for their capability.
The government can't afford these people straight up, so they slap "Presidential" or "White House" on a fellowship to make it more attractive.
And it is; a Presidential or White House fellowship is as much a resume maker as working at a FAANG or an Ivy League degree.