The Hebrew word ruwach is used to designate God’s Spirit in the Old Testament, and sometimes it's also used to designate man’s spirit. However, when God breathed life into Adam, a different word is used, which is neshamah, which means "breath," according to Strong's. This word is used in association with giving life to a human soul. So Genesis 2:7 does not use ruwach, but rather neshamah.
When God created Adam, He breathed into him the “breath of life” and Adam became a living soul (Genesis 2:7). The Hebrew word "nephesh" is the word translated "soul" in Genesis 2:7. The Hebrew word nephesh, although translated as "soul" in some older English Bibles, actually has a meaning closer to "living being"...a living, breathing conscious person, rather than to an immortal soul. According to Genesis 2:7, God did not make a body and put a soul into it like a letter into an envelope of dust; rather he formed man's body from the dust, then, by breathing divine breath into it, he made the body of dust live. In other words, the dust did not embody a soul, but it became a soul—--a whole creature. So, in other words, God gave that body of dust life. Though it's a very poor comparison, it's sort of like the story of Frankenstein, when lightning hit that dead body and it came to life.
1 Corinthians 15:45 says, "And so it is written, The first man Adam was made a living soul; the last Adam was made a quickening spirit." The Greek word for "quickening" is "zoopoieo" (Strong's Number 2227), whose word usage includes "life-giving." The Greek-English New Testament shows the literal translation from the Greek as being, "So also it has been written: became The first man Adam - soul 'a living;' the last Adam - spirit 'a life-giving.'" Matthew Henry's Commentary says, "The first Adam was made a living soul, such a being as ourselves. The second Adam is a quickening Spirit..." The first part of the word 'zoopoieo,' which is zoo, or zoe, means “life” in the sense of resurrection life. As a resurrected person (“spirit”), our Lord will “quicken” people or make alive in the resurrection. The second part, 'poieo,' means “to make,” in the sense of produce, construct, form, or fashion. It is equivalent to the Hebrew ‘asah, which means “to make” or “produce by labour.” ‘Asah is used in reference to the manufacture or construction of any thing. Putting these two parts together, we have the basic sense of zoopoieo, “to make alive with resurrection life,” or “to construct alive with resurrection life.”
1 Corinthians 15:45-49 says that He was resurrected from the dead through the Ruach HaKodesh (Holy Spirit) as the second Adam, and, according to 1 Corinthians 15:20-28, that He will reign on the earth during the Messianic Kingdom until its end when all things return back to the Father.
John 5:26 says, "For as the Father has life in Himself: so hath He given to the Son to have life in Himself."
So the Scriptures make a specific distinction between the spirit that animated the first Adam’s soul, “the breath (neshamah) of life” from the Father, and the “quickening” life-giving Holy Spirit (Ruach HaKodesh) of the Father that animated His Son’s soul.
Jesus was given life as a human, by the Holy Spirit, therefore His life force as a man is God’s Spirit. The Breath of Life – (neshamah) that we inherit from our parents was never intended to give eternal life. The longevity that was imparted to Adam and Eve was actually through their diet, the Tree of Life (Genesis 3:22) [Remember there were two trees, the Tree of the Knowledge of Good and Evil, and the Tree of Life, and they were never told not to eat of the Tree of Life until after the Fall, when the angel guarded it with a flaming sword]. When we receive salvation, we receive the Holy Spirit, which enables our souls to have the same life as Jesus, which is eternal. When God created Adam, He breathed into his nostrils the breath of life (neshamah) and Adam became a living soul. This breath of life was never designed to impart eternal life to its bearer. When Adam and Eve disobeyed Him, they were denied access to the Tree of Life, which was the source of food that maintained their longevity (Genesis 3:22).
In the present ministry of the Ruach HaKodesh (Holy Spirit) by whose indwelling the believer is enabled to live a godly life (Ezekiel 36:26-27), the Holy Spirit is called Ruach HaKodesh, not just Ruach. Ruach HaKodesh means the 'Holy Spirit' or 'Holy Ghost' in Luke 3:16 and 11:13, Ephesians 1:13 and 4:30, 1 Thessalonians 4:8, Titus 3:5, 1 Corinthians 6:9 and Jude 1:20. "Kodesh" is an adjective meaning 'holy.' HaKadosh is a substantive that means "The Holy One," as in Ha-Kadosh baruch hu, or "the Holy One, blessed be He." Ruach HaKadosh would mean "the spirit of the holy one" as in a saint; it is not used for the Holy Spirit. Ruach HaKo'desh means "the holy Spirit" just like har ha-ko'desh means "the holy mountain," and admat ha-ko'desh means "the holy land," and ir ha ko'desh means "the holy city." Ruach Elohim means the spirit of God, found in Matthew 3:16 and 12:28, Romans 8:9,14 and 15:19, 1 Corinthians 2:11,14 and 3:16 and 7:40 and 12:3, Ephesians 4:30, and 1 John 4:2.
From Strong's, the most often-documented word used for "spirit" in the Hebrew Bible is ‘ruwach’ - roo’-akh. Strong (1986: 142). The most common word used in the Hebrew Bible for "soul" is ‘nephesh’ - neh’-fesh. Strong (1986: 105). The most used word for "spirit" in the New Testament is ‘pneuma’ - pnyoo’mah. Strong (1986: 78). The most common world for "soul" in the New Testament is ‘psuche’ - psoo-khay. Strong (1986: 106).
So, in conclusion, the Holy Spirit is not same as the breath of life that God breathed into man.