The problem with the PS2 is that it launched in Japan something like 8 months before the US and Europe so a global sales comparison is a bit skewed. That said, I checked some articles wrote during that time and it seems supply was scarce after all three launches, so I guess the comparison could work just like it does with the delayed european launch of the PS3. I could not find hard numbers in the articles, but the general mood during 2000 was that the Japanese sales were sluggish because of a lack of games and that it got boosted in 2001 thanks to Gran Turismo 3 and FFX. The narrative in 2001 for Europe was a lot more positive with the console being reported as selling three times faster than what the PS1 did...
The official shipment numbers for the PS2 were 1.4 million for Japan alone in the first month (March 2000) and it had reached a total of 10.6 million shipped by the end of March the next year with the american and european launches happening in November. So it probably surged past 6 million global just after the occidental launch, so in about 9 months.