It's saying that if God chooses to destroy you, you can just regenerate yourself back to existence. What an absurd idea.
Indeed, and that's at the heart of it. "Immortality" is nothing to do with having life separate from God, but rather, having the guarantee of life precisely because one's relationship with God is so close that He promises to sustain it forever. Remember, only God has the "source of life".
Creatures can only sustain their own life by God's permission. If He wishes to sustain an individual, they can survive despite lack of food or water, or in the most hostile conditions. Think of Israel in the wilderness, Daniel and his companions in the fire, Jonah in the big fish, and so on. In contrast, without God's permission of life, even the most healthy and well-supported individuals will die - as Adam and Eve discovered (and Satan and the unfaithful angels face in future).
Jesus is immortal, not because of his own inherent ability to let himself live separately from God, but because of his Father's promise and intention never to remove life from his Son or allow him to lose it ever again. This, in turn, is based on the Father's confidence that the Son will never do anything that is deserving of death, due to Jesus having already proved supremely faithful - in other words, he has proved he will never do any act of disloyalty or rebellion that will draw him away from the Father (since drawing away from God is itself what leads to death, as Adam and Eve found).
That reward is also given to the anointed kings and priests in heaven, Christ's brothers, when they receive their crown. (1 Pet 1:4, 1 Cor 15:50-55)
Eventually it will extend to all those described in John 6:53, the ones faithful and loyal enough to be accepted "into God's rest" at the end, who will by then have all proved themselves loyal beyond doubt.