Every living material creature has a soul. A soul is what animates otherwise lifeless matter.
Without a soul, a material creature could not be alive. If it is alive, it has a soul. The question then becomes, what are the differences in the souls.
Aquinas defined 3 types of souls
Vegatiative - The soul of plants and simple life, it provides basic animation, such as metabolism and simple movement
Sensative - The ability to relate and interact with the senses. This is the soul of animals.
Rational - The soul of humans.
ONLY the RATIONAL soul is Immortal. The souls of plants and animals cease to exist at their death.
The soul of humans, being Spiritual, lives beyond the death of the body it animates.
Any teachings on animals not having immortal souls comes from Aquinas, which is all simply theological speculation.
The best answer is that we simply don’t know. I do still hold that the animals will be a part of the new earth someday, not just a big ball of mud (as some have speculated).
While not an infallible teaching, it still is the prevailiant teaching of the Church.
And the teaching does not exclude animals being present in the New Earth, in fact, it almost requires it.
The New Earth is a replenishment of the old Eden. God say fit that animals exist with pre-Fallen Man, there is no reason not to believe that animals will exist with the Risen Man.
But they will most likely be new creations, not ‘Fido’ and ‘Fluffy’ Risen from the dead.