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).
If we are strict with our terminology, God is not a person. God is an infinitely-perfect being. But God is a personal being:
Your concern, however, is valid:
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:
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.