According to St. Thomas Aquinas, all living things have some type of soul, but only human persons have an immortal soul, with free will and intellect (reason).
cf. St. Thomas, Summa, I, 75
[Genesis]
{1:20} Dixit etiam Deus: Producant aquæ reptile animæ viventis, et volatile super terram sub firmamento cæli.
{1:20} And then God said, “Let the waters produce animals with a living soul, and flying creatures above the earth, under the firmament of heaven.”
{1:21} Creavitque Deus cete grandia, et omnem animam viventem atque motabilem, quam produxerant aquæ in species suas, et omne volatile secundum genus suum. Et vidit Deus quod esset bonum.
{1:21} And God created the great sea creatures, and everything with a living soul and the ability to move that the waters produced, according to their species, and all the flying creatures, according to their kind. And God saw that it was good.
{1:24} Dixit quoque Deus: Producat terra animam viventem in genere suo, iumenta, et reptilia, et bestias terræ secundum species suas. Factumque est ita.
{1:24} God also said, “Let the land produce living souls in their kind: cattle, and animals, and wild beasts of the earth, according to their species.” And so it became.
So, according to Genesis, all animals, birds, fish, have souls. Interestingly, plants are not spoken of by Gensis as having souls. However, plants are alive, therefore they have a soul in the most basic sense of a first principle of life. In addition, we should distinguish between the souls of plants versus the souls of animals versus the immortal souls of human persons.