No other non-Nintendo handheld has sold as many units as PSP has or been around as long as PSP has. Even if the relative market share for other handhelds has been close to being the same, none of them has sold 70+ million units. PSP is also successful in how much support it has garnered from publishers (a lot if Japanese, but support nonetheless). No other non-Nintendo handheld has gotteh the number of great games PSP has, I'd even say PSP has a much better game library than GameBoy & GameBoy Advance.
PSP is, in no definition of the word, a failure. Not being #1 != a failure. It's pretty much only because DS was SO successful (the most successful handheld ever, by a large margin) that PSP is sometimes not considered as much of a success as it really is. Had DS been of a more "normal" level of success, PSP wouldn't be so far behind in sales, but it just blows everything out of the water thanks to Brain Training, Nintendogs etc. bringing so many casuals to the device.
Besides, "most successful" is relative to the market. I mean, we can just as easily say Nintendo's most successful handheld ever was the one they put up against a PlayStation that launched when Sony was seen invincible, and roughly a generation ahead in performance... what does that say?
when comparing same generation handhelds, Nintendo vs. others, PSP is by far the most successful. Speaking about relativity would be incredibly stupid when even the closest second to PSP has sold, like, 1/6th of the numbers PSP has. And the difference is even bigger when taking the less successful ones (NGage, NeoGeo Pocket etc. sold in the millions, not even tens of millions).
PSP is the most successful handheld against a Nintendo handheld in every possible way. Fact.