No, it is not a synonym for Christian. Surprised?
Although a true Christian is one of God's elect, God's elect are not necessarily Christians. "The elect" is a term to describe those "chosen by God" before the foundation of the world; therefore, it describes those who are true Christians but it also describes all kinds of other people (e.g. Atheists), and even those who do not even exist yet. Let's look at an example. Jones, although having died a Christian, spent a significant portion of his life as an Atheist; even during those years as an Atheist, he was always one of God's elect. For a more Scriptural example, consider Saul of Tarsus. He was not a Christian while on his campaign to destroy the church, killing and imprisoning followers of Christ, yet we know that he was one of God's elect, chosen before the foundation of the world. He was always one of God's elect, but he was not a Christian until later in his life.
I have an Atheist friend who is viciously opposed to Christianity. I like to tease her sometimes, "What if you are actually one of God's elect who has not yet encountered the day of your salvation?" She does not find that amusing, of course, but the sentiment makes the point fairly clear. And it may caution us to exercise care with unbelievers, for they just might be one of God's own but have not yet been regenerated.