Genesis 3:20 KJV  And Adam called his wife's name Eve; because she was the mother of all living.
Yes, we do have a Father of all living things but that one is not Adam! This then requires an explanation as to why Adam names the woman God had given him ‘Eve’ (Chavah in Hebrew). Notice that he did not name her until after the pair had disobeyed God and received God’s decree? He had earlier named all the animals, for God directed him to do that, but that was before the woman was formed from his rib. Yet Adam did not give the woman a name – until now. What prompted that?
God had just spoken to the serpent saying,
“And enmity I do put between thee and the woman, and between thy seed and her seed: he doth bruise thee – the head, and thou dost bruise him – the heel.” Then, to the woman God says, “Multiplying I multiply thy sorrow and thy conception, in sorrow dost thou bear children, and toward thy husband [is] thy desire, and he doth rule over thee,”
When God finishes speaking to Adam, he then names the woman Eve; “for she hath been mother of all living” (I quote from Young’s Literal Translation of the Bible.) It surely does not need to be said that this is all about her being mother to humanity. Eve was never mother to, or nurturer of, animal or plant life. But I say it anyway, for clarity, because God had already given the green light to procreate (Genesis 1:28).
The past tense is interesting in 3:20. Adam was clearly speaking in faith, trusting in God’s assurance that she would bear children, from which all humanity would result. He would be the literal father of all the literal babies that Eve would give birth to, but there’s a deeper, spiritual meaning here.
Hebrews 12:4-11 brings in God as Father of those who have striven against sin; He disciplines, scourges, even, all those whom He calls ‘sons’. The writer contrasts God’s fatherly disciplining of his children with that meted out by our physical, earthly fathers. We respect “fathers of our flesh” (note the plural in vs. 9 – not one father, as in Adam). How much more should we “be subject to the Father of the spirits, and live?” This is where we need to go back to Eden, bearing that in mind.
Adam and the woman attempted the impossible – to rise to a higher level, to be as God himself, but failed, only succeeding in discovering a lack in their humanity. They lacked something. They felt exposed and felt shame. God had never drawn attention to them being unclothed but after judging that first, failed humanity, he provided covering (through sacrifice), to prepare them for the chill winds and briars outside of Eden. Inside Eden there was no failure, not until the first Adam chose to cast his lot in with his wife instead of abiding by God’s prohibitive command. That is why the Bible puts the responsibility for sin entering the world, and death through sin, on Adam’s shoulders (Romans 5:12). Adam gave rise to the first, failed, humanity. But God already had the last Adam, already viewed as “the Lamb slain from the foundation of the world” (Revelation 13:8), the first hint of which came in Genesis 3:15.
But note that the woman was taken out of the man before he fell. Before Adam transgressed, woman was taken out of him. From the ‘Mother of all living’ would come a seed. Not of man, but of woman. Not a progeny of a failed humanity – another humanity altogether – a new creation. Nor would it be a new thing out of the old. This would come by redemption, not by attempts at self-righteousness (note how the woman blamed the serpent, and the man blamed the woman, which was actually blaming God who had given him her?) The seed of the woman said “Ye must be born again”, a birth of water and of Spirit, from above – being born of God, which makes him our Father, for spiritual begetting comes from God the Father – the Father of spirits, as in Hebrews 12.
This brings me full circle so I will stop, resisting going back to stop at all the myriad other details that could (and should) be brought in along the way to give a fuller answer to your important question.
The name חַוָּה Chavah is a derivative of the Hebrew word חַיָּה chayah meaning “living” in singular feminine form. In the Chabad article “Mother of All Life” - Sara Esther Crispe states :
Chavah embodies both the essence of life itself and the creative ability to grant that life to others. The idea of “mother of all life” expresses not only the ability to physically give birth, but also to create, nourish, and enhance all facets of life. This is the ability of a mother - to take something from the state of potential, develop it, and bring it to actualization through her creative abilities.
[To honor God, The-Man הָֽאָדָם Ha-Adam named אִשְׁתּ֖וֹ his-wife: חַוָּה Chavah whose image was made in the likeness of God & whose creative abilities were like יְהֹוָ֨ה The Father of all Life].
Genesis / Bereishit 3:20
“And The-Man named his wife Chavah, because she was the mother of all life.” ( וַיִּקְרָ֧א הָֽאָדָ֛ם שֵׁ֥ם אִשְׁתּ֖וֹ חַוָּה כִּ֛י הִ֥וא הָֽיְתָ֖ה אֵ֥ם כָּל־חָֽי )