Yes, indeed, the umbilical cord develops from the cells of the embryo. When the germ cells divide, stratification occurs into the inner part, which will become a small person and the outer, which is introduced into the uterine cavity and subsequently forms the placenta and umbilical cord.
The method of fetal stem cells collection from the umbilical cord at the birth of a child has become widespread , for the so-called "biological insurance", since in case of a serious illness, these cells will help save the child's life.
I will slightly supplement the answer. In fact, the placenta is partly made up of maternal tissue and partly from fetal tissue. Spiral arteries from the arteries of the uterus carry blood to the base of the placenta. Then the blood fills the intervillous spaces in which the chorionic villi are immersed. This picture reminds me of a tree with roots and crown (vessels of the fetus in the placenta) on top of which are dense vines of the vineyard (vessels of the mother). Both circulatory systems come close to each other, but the blood does not mix anywhere; the mother's blood from the child's blood is everywhere separated by thin walls of villi.
All cells of a child contain only the child's DNA. Your own navel is a remnant of the umbilical cord - your body does not reject it?
Another answer is that the DNA of the umbilical cord, like the rest of the organism, is 50% identical to that of the mother.