Living creatures with the breath of life have hope of life eternal. Revisit your Concordance and consider the meaning of
nephesh, nephesh chayyah, and
ruach. Note where these words are used, and in relation to which creature. If you can do so and come away with the impression that man alone among creatures with the breath of life is a triune being, you're not reading what I'm reading.
Dogs do have souls. They have a place alongside us in heaven as part of our household, though, regardless of how fine a point you put upon the intended meaning of nephesh, nephesh chayyah, and ruach. But, that is a debate for another day.
I read Hebrew and Greek, so I usually consult OT and NT word studies, such as Kittel. . Nephesh can simply mean breathe or soul, and it can even mean throat. God is the Creator of all living things, they have biological existence (nephesh). Ruah can mean breathe, wind, spirit, etc. Looking in a concordance is not helpful. It is the difference between diachronic and synchronic meanings of words. As in all things, context is king.
The Hebrews were not pet or animal lovers in the sense that many of us are about our pets. They certainly thought that dogs were an abomination. Many kinds of animals were considered unclean. Even shepherds and sheep were thought to be unclean and the bottom of the social ladder. They definitely did not think of animals as having souls or consciences. As far as they were concerned, animals were nothing more than beasts to be eaten, worked, or destroyed. It is in this wider context (Sitz im Leben) that the idea of animals having human souls or eternal life would be most odd. I can find nothing in the OT that would indicate that animals were redeemed by the Lord.
There is an old saying that he who asserts must prove. I do not know of anything in Scripture that supports the view that animals have souls, in this case “dogs.” The idea for a Hebrew that dogs had souls would be horrific to say the least.