"2 John" is a Jewish text concerned with Jewish things and it is an important misstep to assume that he is writing to the Christian ("church age") assembly. We know this because like 1 John he is beholden to the man Jesus:
NIV 2 John 1:
9 Anyone who runs ahead and does not continue in the teaching of
Christ does not have God; whoever continues in the teaching has both
the Father and the Son. 10 If anyone comes to you and does not bring
this teaching, do not take them into your house or welcome them. 11
Anyone who welcomes them shares in their wicked work.
What were the teachings of the earthly Christ? He taught the law of Moses:
NIV Matthew 5:
17“Do not think that I have come to abolish the Law or the Prophets; I
have not come to abolish them but to fulfill them. 18For truly I tell
you, until heaven and earth disappear, not the smallest letter, not
the least stroke of a pen, will by any means disappear from the Law
until everything is accomplished. 19Therefore anyone who sets aside
one of the least of these commands and teaches others accordingly will
be called least in the kingdom of heaven, but whoever practices and
teaches these commands will be called great in the kingdom of heaven.
This is because Jesus was sent only to the Jews because his mission was to be rejected by the Jewish leadership and so open up the way for the gentiles (Romans 9-11) and the end of the age had not yet occurred (it would occur a few years later, in 70 AD).
So 2 John is writing to a collective but it is the collective of the faithful Jews, not of the faithful Christians. These would not be a local assembly but rather the ingathering in Jerusalem and spreading further out. These are those who have believed that Jesus is their messiah and is returning imminently to rescue Israel from Rome, set up David's throne and establish the 1000 year reign of peace and prosperity for Israel (see Zechariah 14).
For the author of 2 John the elect lady is the messianic community. The same was true for Peter:
NIV 1 Peter 1:1 Peter, an apostle of Jesus Christ, To those who are
elect exiles of the Dispersion in Pontus, Galatia, Cappadocia, Asia,
NIV 1 Peter 1:10 Therefore, brothers,g be all the more diligent to
confirm your calling and election, for if you practice these
qualities you will never fall.
James likewise is writing to the messianic communities:
NIV James 1:1 James, a servant of God and of the Lord Jesus Christ,
To the twelve tribes in the Dispersion: Greetings.
And what is his concern? The necessity of works (faith without works is dead, pure religion is visitation and purity).
And so on.
So 2 John is written to the messianic community which is personified as a woman, which we see also in Revelation:
NIV Revelation 12:
1And a great sign appeared in heaven: a woman clothed with the sun,
with the moon under her feet, and on her head a crown of twelve stars.
2She was pregnant and was crying out in birth pains and the agony of
giving birth. 3And another sign appeared in heaven: behold, a great
red dragon, with seven heads and ten horns, and on his heads seven
diadems. 4His tail swept down a third of the stars of heaven and cast
them to the earth. And the dragon stood before the woman who was about
to give birth, so that when she bore her child he might devour it.
5She gave birth to a male child, one who is to rulea all the nations
with a rod of iron, but her child was caught up to God and to his
throne, 6and the woman fled into the wilderness, where she has a place
prepared by God, in which she is to be nourished for 1,260 days.
But Paul was the architect of the new humanity because God committed to him the task of dispensing the mysteries of Christ. And he is not beholden to the earthly, Jewish messiah:
NIV 2 Corinthians 5:
16From now on, therefore, we regard no one according to the flesh.
Even though we once regarded Christ according to the flesh, we regard
him thus no longer. 17Therefore, if anyone is in Christ, he is a new
creation.b The old has passed away; behold, the new has come.