I was first asked this question many years ago. I’ve revisited it Monday and Tuesday in a post for Biologos.org. The question gets at the basic nature of human identity and personality. As people we perceive ourselves as more than our bodies and brains; what we think and believe extends beyond our neurons and synapses. Such an understanding of human identity and personality (sentience, soul or mind or the variety of other words we use to describe this understanding) famously originates with Descartes (“I think therefore I am”) but surely finds earlier origins in places like the Biblical narrative where God cautions Adam and Eve against eating of the tree of knowledge under penalty of death. Since they eat and don’t die physically, right there in the beginning we have a bifurcation of definition (physical and spiritual/relational) if not existence itself.
The good news is that as Christians who believe in the resurrection of the body (coming off Easter), we hold out somehow to an embodied soul even after we die and rot or burn (depending on your choice of burial or cremation). How this plays out physically is only speculation. We will have whatever Paul meant by a “spiritual body” (1 Cor 15) since it will be a body no longer subject to the laws of biology and physics (which will have to have been rewritten).
For now? Yes, a clone would have a soul. But here’s praying we never get to human clones. I think they could end up as tortured souls.