I’ve been researching this a bit more, and it looks like it has something to do with the P3 color gamut. Figma will be set to unmanaged by default which, if you’re on a Mac that supports P3, should display colors in the P3 space, and everything looks more saturated.
Setting to sRGB is what “fixes” this so you see more muted colors like you typically do on the web.
Now, I’m still unsure how this helps solve the Illustrator issue. In the OP’s case, colors in Figma appear saturated, suggesting Figma is set to unmanaged and displaying P3 colors, and would need to be set to sRGB. In your case, this is the other way around.
Another thing I’ve noticed is that grabbing colors from screenshots in Figma is unreliable. When I take a screenshot of anything on my P3 Mac, the image is saved in the RGB space, and when I drop it into Figma, it appears desaturated. In your example, is the image in Figma a screenshot from Illustrator? If it is, I wouldn’t trust it.
Not sure if this gives you a solution, but hopefully some of the info is helpful in finding one. Cheers!