It has been a common Catholic tradition that animals do indeed have souls, animal souls. In traditional theology a soul is a life animating principle thus everything that is “alive” has a kind of soul. The ancient philosophers usually grouped life into three different kinds of souls: vegetative, sensitive, and rational. Humans would have been considered to have rational souls while animals would have sensitive souls.
There is no official Catholic teaching on this matter since Scripture doesn’t say much about it and it has little to no effect on our salvation. Someone could reasonably take a pro or con position on the matter and be well within Church teaching. But, yes, there is a Catholic theological tradition of animals having (sensitive) souls.
