Neoprene rubber finishes
There are lots of variations on neoprene.  It can have different additives that affect characteristics like softness and surface appearance.  Sometimes it is bonded to a fabric.  There are variants that are very soft and a little tacky that will grip a surface for extreme non-slip.  But as long as it's rubbery and soft enough to conform to the surface, it will act as a non-slip material.
An oily residue could be either oil of some kind used in the manufacturing, or an additive that's leaching out.  Many rubber and plastic materials will release some additives, either mostly right after manufacturing or much later, as the material ages.  In some cases, it's designed to work that way to serve a purpose, like acting as a mold release.
Neoprene doesn't receive a &quot;finish&quot;.  Whether the surface is shiny or matte, that's the characteristic of what was made.  You can't change that, or the tackiness, later (although you could apply some temporary adhesive to make it tacky if that's required).
Oily residue, though, can be cleaned off.  It's a contaminant, not part of the neoprene.  There are a few solvents that will damage neoprene, but soapy water or alcohol will be safe for cleaning it.