Atman or soul is only the most subtle part of the human body
Maanushatanoh sukshmatamaamsha atmaa.

(Atman or soul is only the most subtle part of the human body)

In Gita it is said that God enters the human body. This does not mean that God enters the inert human Gross body only. The human body is the human being and is a composite of three bodies (Gross, subtle and causal). The causal part is called as soul or Atman. Hence, when Gita says that God enters the human body, it means that God enters a human being, which is a composite of Gross, subtle and causal bodies.

Veda says that two birds exist on the tree

Sthulavrukshe jivaatmaa suparnah.

(In Vedic statement, the gross body is the tree and Jivaatman, the composite of subtle and causal bodies is said to be the bird).

Veda says that two birds exist on the tree. Here the inert gross body is the tree. The two birds are God and the Jivaatman of the tree. Here the Jivaatman is the composite of subtle and causal bodies, being the owner of the gross body. Here the tree and bird are mentioned as two items. The tree stands for gross body. The bird stands for both subtle and causal bodies, because these two bodies cannot be isolated like water and its wave.

Jiva cannot exist without Atman

Apruthak karanam tayorna kaamya atmaa.

(Jiva and Atman cannot be isolated from each other. Though Atman can exist without Jiva, it is not desirable).

Jiva cannot exist without Atman as the wave cannot exist without water. Hence, when Jiva is mentioned, Atman is simultaneously mentioned. According to Advaita, in a realized soul, the Atman can exist without Jiva. But here also, the qualities of the world are subsided and the water (soul) is without worldly qualities (waves). However, the divine qualities (waves) must exist in the realized soul also. A soul without any quality is not the real salvation and such salvation of monism (Advaita) is only useless inert state of a stone that is obtained in deep sleep every day. Hence, though the soul without any quality can be achieved by effort, such a state is meaningless and hence the soul also should not exist without any quality in salvation. Salvation is only liberation from the worldly qualities and simultaneous achievement of divine qualities.