GoKawiil"It was a big sales dud" is the short answer. Despite initially selling 100,000 units during its launch week back in late 2003, sales quickly tapered off — even after an updated model released in 2004 (which Engadget covered at the time). Though Japanese reports from back then suggest Sony shipped several hundred thousand units by March 2004, the company was reluctant to publish concrete sales data. Over 20 years later, there still aren't reliable lifetime sales figures for the PSX.
Why did the PSX bomb as hard as it did? The somewhat niche selling point of the system didn't help: a hybrid video recorder/DVD player that's also a PS2 is hard to market succinctly. In the spring of 2004, GameSpot reported internal conflicts at Sony might have hindered any potential success the PSX would've had. An unnamed source "close to Sony" told the site, "A lot of things have been said about the failure of the PSX inside the company, but it was probably due to the unrealistic way that its development was conducted." Said source would then claim that Sony's games division and its Blu-ray disc department (who were both handling production of the PSX) "couldn't work too well together," ending up with "a machine that's neither a game console nor a DVD recorder." Cue the machine being phased out in early 2005.
A couple of additional factors sealed the early demise of the PSX. In particular, it cost a heck of a lot more than a PS2. The 160GB model cost ¥79,800, while the 250GB PSX was priced at ¥99,800 (roughly $730 and $910 respectively, circa 2003). By contrast, the PS2 had dropped to around ¥19,800 in Japan the month before the PSX launched, with the colossally successful console costing around $179 in the US by that time.
There were also broadcasting rights issues that made a full-scale international release tricky to pull off. Because the PSX was a DVR designed around Japanese TV, an overseas version of the machine would have required the additional overheard of Sony coordinating with US and European broadcasting regulators.