What does it mean when Christians say that God is a person? (Representative example here.) Does this mean that Christians believe that God is human?
My experience is with Judaism (in Judaism, God is not referred to as a person).
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If we are strict with our terminology, God is not a person. God is an infinitely-perfect being. But God is a personal being:
When we say that God is a personal being we mean that He is intelligent and free and distinct from the created universe. Personality as such expresses perfection, and if human personality as such connotes imperfection, it must be remembered that, as in the case of similar predicates, this connotation is excluded when we attribute personality to God. It is principally by way of opposition to Pantheism that Divine personality is emphasized by the Theistic philosopher. (source)
Your concern, however, is valid:
Yet sometimes men are led by a natural tendency to think and speak of God as if He were a magnified creature — more especially a magnified man — and this is known as anthropomorphism. Thus God is said to see or hear, as if He had physical organs, or to be angry or sorry, as if subject to human passions: and this perfectly legitimate and more or less unavoidable use of metaphor is often quite unfairly alleged to prove that the strictly Infinite is unthinkable and unknowable, and that it is really a finite anthropomorphic God that men worship.
When we refer to God as a person, we mean that He is personal, not that He is a magnified human.
When speaking of God, we usually try to reserve the term "Person" to one of the three Divine Persons of the Trinity. There is one God, who is a Trinity of three Divine Persons: the Father, the Son, and the Holy Spirit. The Son is Jesus Christ, who is true God and true man. When speaking of Jesus being a person, we say that He is a Divine Person, not a human Person, and we say that He is God, in perfect unity with the Father and the Holy Sprit.
That God is referred to as a Person is, first, in contrast to the idea of some that the universe was created by some impersonal force--a force that has no personal being, but just some kind of cosmic energy. Such a creative force would have created something greater than itself--beings with Personhood--so this is not reasonable.
The idea that God is a Person refers to the how He Himself describes Himself in His Word as well. God feels, loves, and relates to His creation. As A.W. Tozer wrote:
God is a Person, and in the deep of His mighty nature He thinks, wills, enjoys, feels, loves, desires and suffers as any other person may.
If God were not a Person, He could not love, as that is something only a Person can do. The doctrine of the Trinity is interesting in this aspect, because God reveals one of His attributes is love. However, if it is an attribute (and not a behavior) of God, it has to exist apart from anything else. As the Father loves the Son, the Son loves the Father; the Father loves the Spirit, etc. Thus, only as a Trinity can God be love from all eternity.
So, what is meant is that God is the source for the personal attributes that we as His creatures share in--that of loving, feeling, knowing, etc. In this we bear the image of God.