The problem is not with the peroxide unless it is a high concentration. For example 3% hydrogen peroxide does very little to tissues when applied. But a 35% peroxide will do some significant burning and 100% hydrogen peroxide can literally set you on fire. I have heard of a case where a truck carrying 100 peroxide, which is actually a thick blue liquid, crashed leaking the peroxide. The peroxide ignited the asphalt. So concentration is a big factor. The small amount of peroxide in the magnesium peroxide is not enough to cause damage. In fact magnesium peroxide is so expensive to begin with and requires hazardous shipping over a certain percentage, which is also expensive that these products will not contain very much. Last time I checked on pricing of magnesium peroxide about 14 years ago the only source I could find wanted about $160.00 a pound plus the hazardous shipping charges. Therefore the magnesium peroxide is diluted with magnesium hydroxide.
And this is where the problem comes in. Magnesium oxide reacts with water to form magnesium hydroxide. But as with all hydroxides, magnesium hydroxide is caustic. Other examples of hydroxides include calcium hydroxide (lime) used in making cement and sodium and potassium hydroxides used in drain cleaners. Containers with any of these ingredients have warning labels on them because tissue contact can lead to chemical burns.