Is God a person?

Nowadays, classical theism is contrasted with theistic personalism. Likewise, many proponents treat Thomism as a virtual synonym for classical theism, although Thomism is just one version of classical theism. Because some young Calvinists are attracted to Thomism, I'd like to say a few things about theistic personalism. I'll use Brian Davies as my frame of reference: An Introduction to the Philosophy of Religion (2004, 3rd. ed.).

…according to classical theism, God is not a person. When we speak of persons, we are normally referring to human beings (8).

That depends on the context. In theological discourse, you have Trinitarian persons, angelic persons, and disembodied souls. 

Human beings have bodies and are parts of a changing and changeable universe (8).

Living human beings have bodies. 

For classical theism, however, God is not an individual belonging to a kind…God is simple means in part that God is not a member of any genus or species. They are claiming that God is not what we would ordinarily call an individual. To call something an individual is usually to imply that there could be another such thing distinct from it though just like it. In this sense, different people are individuals. But in this sense, says the classical theist, God is not an individual. He belongs to no kind or sort (8-9).

This gets to be very dicey. 

i) If monotheism is true, then there's a sense in which God is one of a kind. Sui generis. A class apart.

ii) If Trinitarianism is true, then there's a sense in which the Father, Son, and Spirit are three of a kind. 

iii) Likewise, if monotheism is true, then there's a sense in which God is an individual, in contrast to other individuals (i.e. creatures). 

iv) By the same token, if Trinitarianism is true, then there's a sense in which the Father, Son, and Spirit are distinct (but similar) individuals. 

v) Now, I don't think God is a property instance of a generic divine nature. I don't think the persons of the Trinity are property instances of a generic divine nature. They aren't finite exemplifications of a generic exemplar. Rather, I'd model them on mirror symmetries. Each mirror symmetry is exhaustive. Each mirror symmetry reflects the content of its antipode, in one-to-one correspondence. Yet each presents a unique frame of reference (e.g. chirality). 

According to the teaching that God is simple, God also lacks attributes or properties distinguishable from himself (9).

The real nub of the issue is whether God's attributes are distinguishable from each other.

Yet, what are we to understand by expressions like "person without a body" and "disembodied person"?…Aristotle holds that persons we call people are essentially corporeal…According to [Locke], persons might swap bodies with each other…But it would, he argues, be the same person...The view that persons are not essentially corporeal is most often associated with René Descartes…So the history of philosophy contains examples of authors who take persons to be distinguishable from what is essentially corporeal (10-11).

What's striking about this exposition is how he correlates that position with theistic personalism, in contrast to classical theism. Yet the idea of discarnate persons is standard in Christian theology, from God through angels and demons to disembodied souls (during the intermediate state).

His dichotomy doesn't make much sense unless you correlate classical theism with physicalism. But that's not traditional orthodoxy. 

There is a sense in which God is not "a person". If Trinitarianism is true, then it's more accurate to say that God is personal.

Yet classical theists also typically insist that none of this means that we therefore have a grasp of God or a concept which allows us to say that we understand what God is. This fact partly emerges from the way in which classical theists often characterize God in negative terms…But it also comes out in the fact that classical theists tend to deny that words used to characterize God mean what they do when applied to what is not divine (7).
[Theistic personalists] also sometimes suggest that words (especially adjectives) used by believers when speaking of God are most naturally to be construed in the same way as when they are applied to people. Theists say that God is, for example, knowing, loving, and good. But we know what it means to say that people are knowing, loving, and good. So, reasons many a theistic personalist, we know something of what it means to say that God is knowing, loving, and good (14). 

The way he dichotomizes the two positions makes it hard to distinguish classical theism from pious agnosticism.