OG xbox had more then just a resolution advantage.
What eastman mentioned wasn't what Sony could possibly stuff in the PS2 instead of wonky hw. But what a pc of the time era around the PS2's launch could do if optimized as much for it as was done for the PS2. A late 2000/early 2001 pc wasn't that far off from an OG xbox in terms of capabilities. Xbox would've sported a somewhat more advanced GPU, but the pc a much more capable CPU, RAM (with dedicated pools to cpu and gpu) and a close enough GPU to still blow the ps2 out of the water in almost all regards.
Its capabilities allowed it to stand out in its own terms rather than being another outdated PC that has nothing else to show besides what a PC of its time could push.
And for that very reason some PS2 exclusives were unmatched by PC games for years to come and some PS2 to XBOX or PC were missing effects (or looked different) that heavilly relied on fill rate while PS2 to PC games required significantly higher minimum specifications on PC by large to run at an acceptable performance. See for example Silent Hill 2 and MGS2.
PS2 exclusives being unmatched by pc games 'for years to come' isnt what actually happened though. The PS2 was severly lacking in many areas that cant be over-looked. PC games stood above what the PS2 was capable off in the total image. PS2 was good at effects due to its architecture, and devs worked hard to get around its many limitations, but that didnt lift it above what early 2000's pc hardware did with games.
'PS2 graphics' to this day is still a term, over two decades later. Largely due to it being the most popular console backthen, but also due to the PS2 offering the weakest graphics during the 6th gen (vs Xbox, GC and even the evil pc).
Mipmaps where basically broken, AA didnt really exist, field rendering was a problem even half a decade into the gen, multiplat games both performed and looked worse despite it being the main target platform for most games, and whenever the system was truly pushed (SoTC, MGS3 etc) the framerate tanked (below 15fps for sotc) and environments turned into corridors (MGS3 30fps with a loading screen every small jungle section).
When you want to compare system performance (as you mention ps2 to xbox), you cant look at exclusives or its subsequent ports as they are optimized to one platforms strengths and then ported, like MGS2 to Xbox, which held up quite well, considering. Xbox to PS2 ports though? 'what am i looking at' (splinter cell, and this racing game). Xbox 'exclusives' like Doom 3, HL2, Far Cry, conker, quantum redshift, halo. The PS2 wouldnt really be up to it with these. While i can imagine the Xbox doing quite well, even if not as great, in games like ratched and Zoe2. Some even argued the vertex shaders could do some impressive particle effects work.
And thats where you see the differences.
PS2 was a great, but not directly because of its hardware and architecture. The machine would have done just as well if it had sported a different architecture (and have a stronger machine at it). With the PS2 it seems Sony looked at the PSX and designed from there.
All this you can find on these very forums, theres actual developer discussions who worked on all these platfroms (even gamecube). The PC was more capable, had means to show it but it cost you alot. An Xbox at the time was a very good alternative to the PC as it shared many games (abit like today). That console was a look into the future basically, build in HDD, network/xbox live and x86/'off the shelf hardware'.