best way to remove machining marks, so they are not seen on the finished anodized part

Low rigidity hobby routers were never really intended to give great finishes on metals. Sanding/grinding and polishing are about the only way. Tumbling for non-flat parts. If you're hard anodizing, you might not go to quite as high a grit as you would on a raw finish. For regular anodizing, you'd probably still want to go quite high since the oxide layer is very thin and the surface finish will directly impact it.

The real solution is to eliminate the machining marks at the source as much as possible- you'll always get some swirls no matter what, but they should be a few microns deep, not tens of microns. Better quality tooling, more optimized toolpathing (fully exit the part and deck in one axis instead of cornering in contact, or use circular strategies, or...), better feeds and speeds, higher machine rigidity, better spindle quality... It's a long road to get from structurally acceptable to aesthetically good.