Does this mean that God breathed his spirit into Adam giving him life.
The exact nature of what God blew into man is discussed by some of the commentators.
The Sfrono writes
G’d personally blew a soul of life into man, a life force which had been readied to absorb what was previously called צלם אלוקים, “the image of G’d.” This has been attested to in Job 32,8 ונשמת שדי תבינם,”by means of the breath of the attribute of Shaddai You endowed them with understanding.”
The Chizkuni explains
“He blew into his nostrils a living soul.” G-d personally blew the breath of life into the human being, something He had not done for any of His other creatures. Why was all this necessary? This was in order to enable man to have the wisdom, i.e. holy spirit, to enable him when viewing all the animals to name them correctly after having discerned how each was different from the other.
In his discussion of the nature of the soul, Rabbeinu Bahya writes
The soul of intelligence is something exclusive to man, something he shares with beings in the higher worlds, beings which live forever in their sacred domains The seat of this soul is in man’s brain, and it is generally referred to as נשמה. This is what the Torah described in our verse as the נשמת חיים which G-d “blew” into Adam’s nostrils.
The Radak doesn't limit himself to some notion of soul which is totally separate from God. He talks of a life-force. In fact, he writes
G’d blew into man some of the spirit prevailing in the highest regions on earth...man’s life force is called here נשמה, to alert us to the fact that the origin of this life force is not physical, has not been supplied by earth, in fact could not have been supplied by earth.
While other commentators write at length about the nature of the soul (such as the Ramban) that was blown in, I haven't found one that says explicitly that the soul is some "part" of God that is synonymous with the spirit of God mentioned in 1:1, though it is somehow an image of God. The Alshich spends a bunch of time discussing how whatever is blown in is still connected to God (which would make it identical with some essence still "inside" God) and he speaks of a neshama elyona (a superior soul).
The meaning of Genesis 2:7
Usually the term ruach elohim is translated to mean “the spirit of G-d,” Actually ruach, in this context, means “wind.” Likewise, elohim means powerful because G-d is more powerful than anything else. Thus the correct translation reads “a strong wind,” and not a spirit.
Verses 1:26 states, “Let us make man in our demut. Verse 1:27 relates that “G-d made man in His tzelem, in the tzelem of G-d, He made him.”
The word demut means “likeness.” G-d created man in His “likeness.” But G-d has no body, so man cannot have G-d’s “likeness.” The meaning of tzelem is “shape” or “form.” A divine “shape?” 12th-century sage, Moses Maimonides felt that the two denotes the essence of two things. What is the essence of man? His intelligence.
Genesis 2:7: “The L-rd G-d formed man of the dust of the ground, and breathed into his nostrils the breath of life; and man became a nefesh chayah.” Onkelos does not renders this as “a living being,” but implies the characteristics of man, his ruach m’mal’la, “one with the power of speech,” a phrase denoting a certain level of intelligence. The targumic interpretation and Egyptian Jewish philosopher Philo understood this intellect to be “the breath of life” in his Specialibus Legibus 4:123.
The term ruach Elohim is a wind that hovered over the water. The Bible is exclaiming that “G-d created a wind that blew upon the water.” Similarly, the phrase tzelem Elohim, “the image of G-d,” Elohim means “awesome.” G-d created people with the awesome power of intelligence. Thus, G-d states in Genesis 1:26: “Let us make man in our image, after our likeness”. Maimonides felt that this denotes intelligence. That a person should develop his or her ability to think because that is like G-d. Thus, we are created in the image of G-d. A Divine Spark. G-d also gave man a soul, the neshama.