why on earth blue color adds grey edges to my character? How to fix it?

When Illustrator rasters artwork using antialiasing, there are two options: Type Optimized and Art Optimized. Use Art Optimized, as it renders the artwork in the background in high resolution, then reduces it to the final required resolution. In the Type Optimized mode, every object renders separately from the bottom up at the resolution selected, it renders each  with a partially-transparent anti-aliased edge. Unfortunately, even on objects that butt up against each other, this creates a semi-transparent 1-pixel "seam" that allows whatever color is behind the objects (in your case grey) to show through.

Are you using Art Optimized for the Anti-Aliasing method during export?

Are you exporting to a raster format?

If so are you using Pixel Perfect, and snap to Pixel Grid?

https://helpx.adobe.com/illustrator/how-to/pixel-perfect.html

https://helpx.adobe.com/illustrator/using/pixel-perfect.html