First, I think that "fellowship" is an extremely weak translation of the underlying Greek word - κοινωνία. A much better English word to use would be the one which retains the Greek as its root - communion. Consider the translation of these verses from the Holy Apostles Convent Orthodox New Testament:
That which we have seen and heard we relate to you, in order that you may have communion with us; and our communion also is with the Father
and with His Son, Jesus Christ.
When one understands that John is speaking of union with and not mere "fellowship" with God, a different meaning emerges from the text. When we are united with others who are united with God, the relationship is distinct from communion with God in that we are united with distinct persons, but indistinct in that we are commonly united with God.
Orthodox Christian theologian and monk Justin Popovic explained v.3:
Living in communion with the Lord Christ, having communion (κοινωνίαν)
with Him is, in fact, to be located in eternal life. That life, in
actuality, is communion with the Trinitarian Godhead. To commune with
the Trinity, through the God-man, and to come to know the Trinity is
nothing other than living eternal life. Then man feels and sees
eternal life in himself. Also, he is located in a community of
eternal human beings.
Commentary on the Epistles of St. John the Theologian (tr.
Sebastian Press, 2009), p.14
Regarding your second question, "Is the writer trying to use Christian fellowship [κοινωνία] to also make the point that the Father and His Son Jesus Christ is God?", I think the answer is no, not in that particular verse. He seems to make a much stronger case in the opening of the Epistle. These verses speak of the Word of the Life, although the King James and other versions drop the definite article before "Life". Athanasius discussed these verses in his Discources Against the Arians:
But that the Son has no beginning of being, but before He was made man
was ever with the Father, John makes clear in his first Epistle,
writing thus: ‘That which was from the beginning, which we have heard,
which we have seen with our eyes, which we have looked upon, and our
hands have handled of the Word of Life; and the Life was manifested,
and we have seen it; and we bear witness and declare unto you that
Eternal Life, which was with the Father, and was manifested unto us.’
While he says here that ‘the Life,’ not ‘became,’ but ‘was with the
Father,’ in the end of his Epistle he says the Son is the Life,
writing, ‘And we are in Him that is True, even in His Son, Jesus
Christ; this is the True God and Eternal Life.’ [5:20] But if the Son
is the Life, and the Life was with the Father, and if the Son was with
the Father, and the same Evangelist says, ‘And the Word was with God,’
[John 1:1] the Son must be the Word, which is ever with the Father.
And as the ‘Son’ is ‘Word,’ so ‘God’ must be ‘the Father.’ Moreover,
the Son, according to John, is not merely ‘God’ but ‘True God;’ for
according to the same Evangelist, ‘And the Word was God;’ and the Son
said, ‘I am the Life.’ [John 14:6] Therefore the Son is the Word and
Life which is with the Father. And again, what is said in the same
John, ‘The Only-begotten Son which is in the bosom of the Father,’
[John 1:18] shews that the Son was ever. For whom John calls Son, Him
David mentions in the Psalm as God’s Hand, saying, ‘Why stretchest
Thou not forth Thy Right Hand out of Thy bosom?’ [Psalm 73:11 LXX]
Therefore if the Hand is in
the bosom, and the Son in the bosom, the Son will be the Hand, and the
Hand will be the Son, through whom the Father made all things; for it
is written, ‘Thy Hand made all these things,’ [Isaiah 66:2] and ‘He
led out His people with His Hand;’ [Deuteronomy 7:8] therefore through
the Son. And if ‘this is the changing of the Right Hand of the Most
Highest,’ [Psalm 76:10 LXX] and again, ‘Unto the end, concerning the
things that shall be changed, a song for My Well-beloved;’ [Psalm 44
LXX] the Well-beloved then is the Hand that was changed; concerning
whom the Divine Voice also says, ‘This is My Beloved Son.’ This ‘My
Hand’ then is equivalent to ‘This My Son.’
Discourses Against the Arians, Fourth Discourse, Chapter 26
