"Incarnation" is from the Latin word, carnis, which means "meat or flesh." To be incarnated is to be embodied, enfleshed.
Many Eastern and New Age religions share the Hindu view of reincarnation, namely that the very same principle of life, jiva, can animate ("bring to life") any imaginable form of life. The same jiva could be incarnated as an insect, a worm, a monkey, an elephant, a horse, a man, a woman, or any other kind of living being.
Hindus believe (as a general rule--there is no Pope, creed, dogma, or catechism for all Hindus) that we will continue to be reincarnated in this universe until we become divine. If we live our life well, the law of karma will reward us by causing us to be reincarnated in a superior form of life; if we live our life poorly, the law of karma will punish us by causing us to be reincarnated in an inferior form of life.
The appeal of this idea is obvious. It takes the pressure off of us for any one lifetime. We don't have to worry about ourselves or those whom we love. If we don't get it right in this lifetime, there are infinitely many more lifetimes available for us to learn our lessons and reach moksha, liberation from the cycle of reincarnation.
The movie, "Groundhog Day," was written to illustrate the doctrines of reincarnation and karma.
The Catholic Understanding of Life and Death
The Soul is the Form of the Body
The Hindu (and, I think, Platonic) idea of the soul is that it is just a passenger along for the ride in a bodily vehicle. Plato said, "Soma sema": The body (soma) is a prison (sema). Just as drivers can get out of a wrecked car and climb into a new one of any type, so Hindus imagine that the soul escapes from one dead body and can take possession of any other type of living being.
The Aristotelian, Thomist, and Catholic philosophical tradition sees the soul as the form of the body. Every living thing, even plants, have souls. It is the soul that causes one living being to be a plant, another to be an animal, and still another to be a human being. Souls are not interchangeable parts. A plant soul always produces a plant; an animal soul always produces an animal; a human soul always produces a human being.
Incidentally, this shows what nonsense there is in the Harry Potter books about breaking souls into parts and preserving those parts alive in horcruxes. A body has parts and we can live if we lose some parts of our bodies, but a soul is simple and indivisible. To live is to be a body informed (inwardly formed) by a soul; when the soul is separated from the body, the living being dies.
On these grounds, it is also nonsensical to think that the same human soul could animate other forms of life. My soul causes me to be me, both in this present body (somewhat worse for the wear) and in my resurrected body in Heaven.
We go through life on earth just once
"Death is the end of man's earthly pilgrimage, of the time of grace and mercy which God offers him so as to work out his earthly life in keeping with the divine plan, and to decide his ultimate destiny. When 'the single course of our earthly life' is completed (LG 48 § 3), we shall not return to other earthly lives: 'It is appointed for men to die once' (Heb 9:27). There is no 'reincarnation' after death" (CCC, 1013).