This may seem like an abrupt change in subject (leading some to believe these verses are a later interpolation) but it makes perfect sense that Jesus, having announced his impending departure, would give his disciples instructions for how they were to behave while he is away. It is their own lack of attention to Jesus’ commands (they’re still shocked by Jesus’ statements about his departure) that leads Peter to say what the others are thinking and redirect the conversation back to Jesus’ absence (ch. 14f.).
The expression “new commandment” is the heart of this verse and it stands in an emphatic position in the original language. John uses this same wording in his epistles (1 John 2:8) so it is clear that John not only took this lesson to heart but also recognized its significance. The “one another” makes clear that this is not a statement about the behavior of believers in the world (though they should certainly strive to be loving, kind, and gracious rather than hateful, rude, and difficult) but is the tone and tenor of life among believers (“the marching order for the newly gathering messianic community” Carson 484).
It is new not because it has not been said before (cf. the two great commandments in Mark 12:28-33) but because this command is based on a new standard (“as I have loved you”) and because it grows out of the new order of things in which God’s people are transformed from the inside out. The law is no longer outside of us but written in our hearts. Our relationship then to one another should mimic and exemplify the relationship between the Father and the Son.
This love and this love alone marks a community of faith apart as true followers of Jesus Christ (v. 35). This is why John makes much of this in his epistle.
John’s Gospel teaches both love for the world (3:16; cf. 20:21) because Jesus is the Savior of the world (4:42) but also love for the community so this love is focused differently but the two are hardly mutually exclusive. The church is laboratory in which we learn to love.
We look at different marks than Jesus does. We look at doctrinal statements and how they observe the ordinances or what group they belong to formally or informally. While those things certainly may be important, those are not the standards that Jesus and the apostles hold out.
Jesus wants to know, do we seek out the lost to tell them the good news of Jesus Christ? Do we love one another the same way the Father loves the Son? Do we seek to love God more than we love the world and more the desires of our own flesh? Do we seek to obey him without hypocrisy but in genuine fear grounded in love? Do we exemplify the character and nature of God in our dealings with others both in and out of the church or the community of faith?
These things are what Jesus holds forth as important and these things ought to characterize us as Christians and as a community of faith, as a church. If we are not like this and do not strive to be like this by His power, then we have no right to call ourselves either.