Honing and sharpening are two different things. When you sharpen a blade, you are removing material (metal) from the blade to create a new, thin edge. Sharpening leaves burrs on the edge, while use of a properly sharpened blade tends to round over the very thin edge. Honing the blade removes burrs and can straighten the edge.
Removing the burrs allows for a smooth cut, while straightening the edge puts the sharp part where it's needed. But in neither case is the honing actually sharpening the blade.
So, no…a honing rod cannot be used to sharpen your peeler. Note that the difference isn't the type of peeler, but rather the tool applied to it. If you want to sharpen the peeler blade, you need to use a tool that will remove material from the blade to restore a thin edge for cutting.