There is a big difference between the animal's soul and man's soul. The animal's soul is the moving force of the body; once it departs by death, it becomes a lifeless body, which is a type of life. Plants and trees also possess a type of life but it is not called a soul; rather, it is called a life that flows into its parts through water without which it withers and falls.
{...We created from water every living thing. Will they not then believe?} [276] Surat al-Anbiyā’: 30.
However, it is not like man's soul that was attributed to Allah as a symbol of honor. None knows about its essence except Allah, and it is only for man. Man's soul is a divine matter and he is not required to comprehend its true essence. It is a combination of the moving force of the body along with the intellectual force (the mind), perception, knowledge, and faith and this is what makes it distinct from the animals' soul.