Animals only have souls, not spirits.
Spirit was specially given to man by God when he created man: “The Lord God formed man from the dust of the ground, and breathed into his nostrils the breath of life, and he became a living spirit, and his name was Adam. (Genesis 2:7). “God Himself is Spirit (John 4:24) and dwells in the universe (Psalm 104:1-4). The human spirit refers to conscience (the innate moral instinct), intuition (spiritual awareness), and communication (matters related to worship. God wants people to "worship Him in spirit and truth" - John 4:23-24).