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.