Sorry, there doesn’t seem to be any valid explanation of the animal prefix.
A botanist of the seventeenth century called the European plant the dogberry tree, because of its dark purple berries, but that throws no light on the “dog” element.
One writer only gives this explanation:
“It is called Dogwood, because a decoction of its leaves was used to wash dogs, to free them from vermin,” but that was written in 1838 and no previous writer seems ever to have heard of such usage.
In this we are inclined to agree with the statement in the Century Dictionary:
“In this, as well as . . . in similar popular names of plants, it is not necessary to assume a definite intention in the use of the animal name.”