Paul sometimes uses "grace" as a broad catchall term to declare the way God acts toward His converted but still occasionally sinful children. In every case, whether referring to a singular gift or a continuing package of gifts that result in salvation, grace must always be perceived as unearned. Here, "grace" is used as a kind of shorthand for the entire ministry of Jesus Christ through which we are given salvation.
Notice that Paul exclaims, "Grace has appeared," just as the manna, cloud, and fire appeared to illustrate God's faithful presence to the Israelites through the entirety of their pilgrimage. Thus God is shown freely providing them with guidance, daily sustenance, and security. Recall that in John 14:18 that Jesus says in relation to giving the Spirit of truth, "I will not leave you orphans; I will come to you." Paul is implying in Titus 2 that Jesus is following the pattern that He established with Israel for the church's benefit.
Paul also describes Christ as the personification of grace, salvation, redemption, teaching, hope, and the instruction and inspiration to live godly lives of overcoming and good works. All of these are shown as aspects of one huge gift that is continuously flowing in our lives.
Even as Paul describes Jesus as the personification of grace, he also uses Him as a synonym for grace and all of its powers and benefits, as though Christ exemplified all aspects of grace rolled up in one package. In this way, we can more easily identify and understand it and its meaning to us. Notice further what Jesus—grace—is doing: It is teaching us. Teaching represents the empowerment of knowledge, wisdom, understanding, inspiration, and discernment regarding our responsibilities. It also helps us to identify the subtleties of Satan's devious, anti-God systems.
We should not make the mistake of thinking of grace as an entity; it is not a "thing" God dispenses. "Grace" is a term that represents the freeness of God's personal, patient, and concerned generosity—His blessings and saving acts that are continuously flowing on our behalf to assist us along the way.
God's saving work in us is not merely an extending of life to everlasting life. It is a creative labor on His part, forming us into the image of Jesus Christ, that requires our freely given cooperation for it to succeed. One of our major problems in fulfilling this responsibility by faith is to think about Him consistently, seeking for and acknowledging His benefits, and then returning thanks and praise to Him for His forgiving, patient generosity.