All living creatures – angels, humans, animals, plants – have souls. However, there are two kinds of souls: spiritual and material. Angels and humans have spiritual souls. Animals and plants have material souls. Something that is spirit is not subject to the decay of change. Thus, angels do not die and human souls survive the separation of body and soul at death. Material things however are subject to change, thus they are subject to decay and death. As animals and plants have material souls, at death their souls cease to exist.
Whether God will choose to re-create animal and plant souls at the end of time when the material universe is transformed (cf. CCC 1060) is something we do not yet know. However, the Church does not teach that animal and plant souls are “disposable” in the sense that they are of little consequence and can be wasted if humans wish to do so. Animals and plants do not have inherent rights, but humans, because they are appointed by God to be just stewards of nature, have a responsibility to be kind to animals and prudent in using the world’s natural resources. See the Catechism of the Catholic Church:
The seventh commandment enjoins respect for the integrity of creation. Animals, like plants and inanimate beings, are by nature destined for the common good of past, present, and future humanity. Use of the mineral, vegetable, and animal resources of the universe cannot be divorced from respect for moral imperatives. Man’s dominion over inanimate and other living beings granted by the Creator is not absolute; it is limited by concern for the quality of life of his neighbor, including generations to come; it requires a religious respect for the integrity of creation (CCC 2415).
What Spirits Are and What They Aren’t by Frank Sheed