We aren't told one way or another on that, but I have no reason to believe you are wrong.
That was in response to...
"Animals souls are generic, instinctive, and do not have the potential for everlasting life."
In Genesis One God created certain things "bara." The Hebrew word means that unique to God, He created 'out from nothing.' In Genesis Two we no longer see God creating out from nothing.
One thing needing to be seen. Bodies for souls were supplied in Genesis Two. They were not created out from nothing. Instead, they were.. "jatsar." Which means? 'Molded and formed" out from something that had already been created out from nothing.
Bodies for man and land animals were not given in Genesis One. The souls of both land animals and man were "bara." Created out from nothing.
It says in Genesis One that "God saw." We do not see that wording in Genesis Two. For in chapter One God in His omniscience saw the whole picture of what was being created in sequences.
Genesis One: Land animal souls were first created. Then along came man. But, man was uniquely created in God's Image. That speaks of our soul.
The animal and human bodies consisting of the elements of the earth that originally were molded and formed by the Lord. There was no "bara" taking place with the bodies. Look at a multi mineral complex on a jar and you will see that out bodies need elements of the earth to be replenished.
In Genesis Two those bodies were not created out from nothing. Both animal's and Adam's body were 'molded and formed' out from the ground.
We are always told how the Lord made Adams body out from the elements of the earth. But, it also says the same thing happened for the animals that live on the surface of the earth.
The bodies for the animal souls God had created "out from nothing" in Genesis One, were provided in the next chapter. We do not find God creating out from nothing any longer once Genesis One ends. It actually says that God "rested" from creating out from nothing.
"Now the Lord God had (jatsar) formed out of the ground
all the wild animals and all the birds in the sky. He brought
them to the man to see what he would name them; and
whatever the man called each living creature, that was its
name. So the man gave names to all the livestock, the birds
in the sky and all the wild animals." Gen 2:19-20
Animal bodies were designed to live and function according to how the earth was created. Same goes for the human body living on earth.
The body is an "organic biological machine" that is the physical manifestation of the invisible soul that animates that body. No need for odd differences between mans and animals bodies, unless God creates special attributes into a creature that He wishes to reveal something about His creative ability.
Unless we cut through the ambiguous wording we see in many English translations this factor found in the exacting Hebrew will often times be missed. Some Bible readers have even assumed that there were two different creations with Genesis One and Two. Not so. Not same things taking place in each chapter. The Hebrew meaning of words holds the key.