It seems to me that the point of abiding in something or with something isn’t just a matter of living together, but it has an element of “staying” attached to it. To live and keep on living together. Abiding is the opposite of simply visiting.
When Jesus came as Immanuel He wasn’t just coming to be with us, but He was beginning an age where God would stay with man. Where sin would not have a foothold to separate us from Him. Jesus marked an era were God would abide with man continually, but also man is called to abide with God.
Jesus talks a lot about this in John 15. The fruit of the spirit of God, which is love, joy peace, patience, etc, can only be had by abiding with God, the true vine. We cannot function in a kingdom capacity unless we are staying with God.
We must be careful to abide and bear fruit. Jesus was very clear with a certain fig tree what the consequences are of not bearing fruit. It’s interesting to me that this is different from the gift of the spirit.
In Matthew 7 there is an account of people prophesying, casting out demons and healing the sick, but when it came down to actually abiding with Christ, Jesus said He never knew them. When you abide with Christ you produce fruit. When you don’t abide with Him you’re cast out (and burned - John15:6.) There’s no room for separation in an abiding lifestyle.
Jesus coming to the earth and being sacrificed as the spotless lamb ripped the veil of the temple from top to bottom signifying the importance and significance of God coming to abide with his people. A restoration of the abiding lifestyle that was intended for us in the beginning is the entire goal of the cross.