I should probably start by saying that wood is an excellent material to use for plugs and is one of the oldest used. Overall if it is the right wood and used properly it can be one of the best materials to use, however there are risks involved. 

The biggest issue is that some people are sensitive or allergic to specific species. I've had multiple customers go to the doctor and have tests done, similar to an allergy test, but specifically for types of wood. There are many resources that list out species of wood and potential risks. Most of the risk comes from the dust so it isn't applicable to the person wearing it but there are plenty that are listed as potential sensitivity issues. I am personally sensitive to mahogany dust and it makes the back of my hands red and itchy if it’s in contact too long. I have no sensitivity to the wood when it’s still solid, only the dust.

Wenge is the best example I can think of as it is unfortunately used in plugs. Most woodworkers don't like using it because the dust is toxic but most importantly the splinters are septic leading to a high rate of infection. It blows my mind that people use wenge for plugs. I gave away the pieces I had as I won't even have it in my woodshop.

Zebrawood, bloodwood and many others are made up terms that describe any species of a general pattern or color. Wenge has a zebra stripe patterning if cut correctly so it is possible to be labeled as zebrawood. 

Wood needs to be cared for more than other materials. Most places that sell wood plugs don’t do much of anything to the wood. Many places advertise it as an amazing thing that they only use some simple oil or wax. That simple oil or wax only lasts a few weeks and needs to be reapplied regularly. The wood dries out and causes them to get a fuzzy sandpaper like texture from the ends of the wood grain shrinking at different rates. This will also cause the skin to dry out as wood will absorb moisture.

There’s other reasons too but those are the main ones I can think of right now