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I don't think it's a PCSX2 thing. I seem to remember real PS2 games lacking colors, but I haven't played many.

Did Sony force developers to use a palette limited to "gray" and "slightly less gray"? Is it a hardware limitation?

I played Shadow Hearts and the colors stood out in how samey they were, now I'm playing Front Mission 4 and it's the same. You can customize your mech's colors from hundreds of choices, and all of them look grey to me. It's repulsive.

Here's screenshots: https://imgur.com/a/p40fUTB

This is the brightest green I could find, and the brightest blue, do they look green or blue to you? The other green and blue choices are far worse.

If it's not colorful enough for you
PCSX2=>Config=>Video(GS)=>Plugin Settings=>click on "Shader Configuration=>and do your changes
I increase the saturation by 15

Don't think Sony forced anyone to use any particular colors. Also 2 games isn't really much of a sample size. One of them is arguably a war game based in a dystopian future so I'd argue it's more to do with art direction than any limitation. I can think of quite a few rather colorful games on PS2, Parappa comes to mind. Maybe even Sly. Those aren't particularly grey looking.

There's some limitation on the GS for sure but it's mostly to do with color depth not with how grey something looks.