Does the wall need to be repainted?

You cannot touch up dark color without it being visible. And when touching up, always dab, never brush.

It also always depends on the color, for me repainting with Caparol worked flawlessly on greasy children's hands + chocolate croissants. Sanding probably won't help much, I would generously paint over the stains again with a small roller.

Do not just sand the grease. It must be degreased. For this, there is methylated spirits or nitro or acetone.