[According to the Scriptures and the Church Fathers the "The People of God" is exclusively the Jewish people, the chosen people of the Old Testament.]
"Church" (Ecclesia) is the designation of the new communion, convoked by Christ, which includes both an eschatological and a cultic [worship] element.
"People of God", therefore, does not directly signify the Church of Jesus Christ, but the people of Israel, the first phase of the history of salvation. It would be the result of a Christological transpostion, or, as we might also say, by an pneumatological interpretation that it comes to refer to the Church...
If descending from Abraham and remaining in the Law of Moses is what united men who formed a people, thus, a blood communion and the ordering of a life in common established by God is what constituted the essential content of the concept, so now it is communion with Christ communicated by the Holy Spirit that makes us "sons of Abraham" and configures us to the divine lifestyle. Men come to be people of God thanks to communion with Christ in the Holy Spirit.
We participate in this communion through the sacraments of baptism and the eucharist, which makes us "one" with Christ (Gal. 3:28). I have summarized it with a formula:
The Church is the people of God only and through the body of Christ...It is the action of Christ in us which allows us to pass from being no-people to being people...
...The doctrine of Saint Augustine on the body of Christ is eucharistically centered. From the body of Christ, which is given to them in the eucharist, the Christians themselves become one, they become the body of the Lord. The contrast between institution and "mysticism" here does not appear. The eucharistic celebration is both: institution and "mysticism". It is an concrete ecclesial event, which not only gathers men in one place, but which leads them in all places to the community with the risen Lord, it makes them one interiorly and exteriorly. The doctrine of Saint Augustine on the body of Christ shows itself to be a eucharistic ecclesiology.
Joseph Ratzinger 24 July 1992 prologue to the new edition of his first (prize winning) doctoral dissertation (1951 [completing the doctorate in July 1953 {Milestones, 102}]): People and House of God in Saint Augustine's doctrine on the Church. (Obras Completas I, BAC: Madrid, 2014, 14-15, 18).
N.B. The expression "Ecclesia de Eucharistia" says all of this very economically. The Eucharist is the source of the new people of God, mysterium Corpus Domini (which refers first the the Most Blessed Sacrament [Ratzinger notes that this term was first used in the twelfth century, in reference to the Holy Eucharist] and then, consequently to those who worthily partake of it). Cf. John 6, and Mystici Corporis.
Furthermore, the word "Christmas": etymologically means the same thing, "Christ's Mass". That is why the devils of the world hate to say it, because it implies the entire truth of the Faith: Incarnation, The [Catholic] Church, Priesthood, Eucharist, New People of God.
Happy ninth day of Christmas!