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The short answer is no, animals do not only not remain conscious after death, they are wholly annihilated.
The longer answer is to correct a slight misconception in your question. It is true that mankind and beasts share both material bodies and the "breath of life" (some would call this a soul), but that does not mean that there is no metaphysical difference between man and beast. In particular, man is made in the Image of God (Genesis 1:26), whereas beasts are not. Mankind, therefore, has a moral component and intellectual component to his life and his nature and his soul, which parts beasts lack. Animals do not have intellectual activity (they don't reason as men reason; they can't do abstract logics, mathematics, philosophy, etc. They cannot think about God). Animals also have no moral activity (whatever an animal does is neither right nor wrong, it is only what the animal does by instinct. They cannot choose the good).
This Image of God is the portion of mankind's life which persists after death. This Image is either "dead," if the person is spiritually dead, is not a friend of God, and has not received the salvation offered us by Christ our Savior, or it is "alive" if the person is living a Grace-infused life and following the way of Christ and Christ's Saints.
The Image of God, indeed, is an immaterial soul. God Himself is immaterial, so it is fitting to call this sort of soul the Image of God. He is intellectual, moral, etc, all the aspects of uniquely human life which the immaterial human soul infuses, and which beasts lack.
Both Protestant and Catholic sources would affirm this understanding of the meaning of the phrase from Genesis, "image and likeness of God."