The Bible teaches that the apostles planted local churches, organized regional-city churches (Titus 1:5), and met as a general assembly to hear appeals from the latter (Acts 15). These churches had officers (Phil. 1:1), with biblically restrained authority (1 Cor. 4:6) over specified members “allotted to” their “charge” (1 Peter 5:3; Heb. 13:17). When one translates that into modern practice, the result is what we ordinarily call a Presbyterian denomination.
Of course, what the author is describing is the worldwide catholic church, with local congregations, regional gatherings, and an ecumenical council. And he calls it "a Presbyterian denomination."
Yes, exactly. The entire church functioned as one denomination. That is how presbyterianism works. So what does that say about our modern Reformed world in which so many try to say that the "visible church" is made up of a bunch of independent evangelical denominations? This is to turn the one Body of Christ into many Bodies of Christ, which is contrary both to the Scriptures and to historic Reformed presbyterianism.
For more, see here and in general here.