This presupposes we understand what a soul is. Biblically it seems that the soul is the seat of emotions, and animals, esp. dogs, can illustrate that, but more significant is they also manifest a conscience. None of this need be unBiblical, as
(Job 12:7-10) “But ask now the beasts, and they shall teach thee; and the fowls of the air, and they shall tell thee: {8} Or speak to the earth, and it shall teach thee: and the fishes of the sea shall declare unto thee. {9} Who knoweth not in all these that the hand of the LORD hath wrought this? {10} In whose hand is the soul of every living thing, and the breath of all mankind.”
However, while “soul” and spirit” are sometimes not translated as distinct, a different word is used in the fullness of revelation. (1Ths. 5:23) It is the spirit of man that is seen as giving man his innate sense of God, and responsibility to Him, and thus man is religious by nature, with worship being an activity peculiar to man. No animal as far i know has built his version of Stonehenge for worship.
Some Christians believe animals will live forever, but the problem with that is where do you stop. Redeemed cockroaches?
They sing praises to God, and obey. Do I need to cite chapter and verse?