In the terms you describe, most - if not all - forms of Christianity are Monotheistic in nature. That said, there are three key concepts (that again most forms of Christianity subscribe to) that are somewhat comparable to 'Shivoham' as you've described it: God's Immanence, the Indwelling of the Holy Spirit in a believer, the Glorification of believers.
Before describing those three concepts, I will firstly explain why Christianity* should be regarded as Monotheistic (versus Pantheistic) in character because God is transcendant over His creation:
...[this] means that God is above, other than, and distinct from all he has made - he transcends it all. - Theopedia.com
This is repeatedly implied in the Bible through descriptions of God as Holy (cf. Isaiah 6:3,Revelation 4:8,Leviticus 20:26) - ie separated, other, above - as well as stated more explicitly in certain passages such as:
8 “For my thoughts are not your thoughts,
neither are your ways my ways,”
declares the Lord.
9 “As the heavens are higher than the earth,
so are my ways higher than your ways
and my thoughts than your thoughts. - Isaiah 55:8-9 NIV
The concept of the transendence of God in Christianity is antithetical to the concept of Shivoham.
Moving on to briefly outline the three more comparable concepts:
Firstly, The doctrine of the Immanence of God means:
..."to be within" or "near" in relation to God's creation. Immanence is closely related to God's omnipresence, in that God is always present within the universe, though distinct from it. God is 'within' the universe in that God is its sustaining cause. - Theopedia.com
This teaching empasizes the closeness of God to his creation and can be seen in passages such as:
27 God did this so that they would seek him and perhaps reach out for him and find him, though he is not far from any one of us. 28 ‘For in him we live and move and have our being.’ As some of your own poets have said, ‘We are his offspring.’ - Acts 17:27-28 NIV (emphasis added)
While this doctrine does not go to the extant of Shivoham in identifying God as inseparable from creation, it qualifies our understanding of transendence - God may be 'other', but He is not completely separated from His creation, rather He is close and His presence sustains it. This is an explicit disavowal of Deism.
Secondly, the doctrine of the Indwelling of the Holy Spirit in believers is inextricably linked to the concept of sanctification and teaches that not only is God 'close' in a particular way to them, but through the agency of the Holy Spirit, he allows them to become partakers in the divine nature:
Through these he has given us his very great and precious promises, so that through them you may participate in the divine nature, having escaped the corruption in the world caused by evil desires. - 2 Peter 1:4 NIV (emphasis added)
Most forms of Christianity are not universalist however, and this particular identification of the (renewed) nature of the believer with the nature of God is restricted to the 'household of faith'.
Finally, the doctrine(s) of the Glorification of the believer address the end stage of spiritual development and while appearing in somewhat different forms in the various branches of those who identify as Christian, most will affirm something very similar to:
...“glorification” is God's final removal of sin from the life of the saints (i.e., everyone who is saved) in the eternal state (Romans 8:18; 2 Corinthians 4:17). At Christ’s coming, the glory of God (Romans 5:2) – His honor, praise, majesty, and holiness-- will be realized in us; instead of being mortals burdened with sin nature, we will be changed into holy immortals with direct and unhindered access to God’s presence, and we will enjoy holy commune with Him throughout eternity. In considering glorification, we should focus on Christ, for He is every Christian’s “blessed hope”; also, we may consider final glorification as the culmination of sanctification.
Final glorification must await the manifestation of the glory of our great God and Savior Jesus Christ (Titus 2:13; 1 Timothy 6:14). Until He returns, we are burdened with sin, and our spiritual vision is distorted because of the curse. “For now we see in a mirror dimly, but then face to face. Now I know in part; then I shall know fully, even as I have been fully known” (1 Corinthians 13:12). Every day, we should be diligent by the Spirit to put to death what is earthly in us (Romans 8:13). - gotquestions.org
In conclusion, while there are some elements of the doctrines of Immanence, Indwelling of the Holy Spirit and Glorification that are somewhat comparable to the concept of Shivoham as described, they cannot be considered as identical, as they assume the transendance of God which explicitly denies it.
*Generally on this site, we seek to avoid characterising 'Christianity' as a whole, as there are many religions that claim such a label that possess exclusionary elements to each other. This is so even within the largest generally recognized meta grouping of Nicene Christianity (Catholics, Eastern Orthodox and Protestants). That said, 99.9% of Christian sects can be accurately characterised as monotheistic despite the very real distinctions in many other aspects of their theologies.