There is a minor mistake in the explanation. You are not the body. The one changing bodies is the atma. You are the soul (atma). One should identify themselves as the soul which is non different and not the body.
na tv evāhaḿ jātu nāsaḿ na tvaḿ neme janādhipāḥ
na caiva na bhaviṣyāmaḥ sarve vayam ataḥ param Bhagavad Gita 2.12
Never was there a time when I did not exist, nor you, nor all these kings; nor in the future shall any of us cease to be.
Here Sri Krishna says all have existed forever. When he says that, he is not referring to the physical body but to the atma which is eternal.
dehino ‘smin yathā dehe
kaumāraḿ yauvanaḿ jarā
dhīras tatra na muhyati 2.13
As the embodied soul continuously passes, in this body, from boyhood to youth to old age, the soul similarly passes into another body at death. A sober person is not bewildered by such a change.
vasamsi jirnani yatha vihaya
navani grhnati naro ’parani
tatha sarirani vihaya jirnany
anyani samyati navani dehi (Bhagavad Gita 2.22)
As a person puts on new garments, giving up old ones, the soul similarly accepts new material bodies, giving up the old and useless ones.
In the Second chapter of Bhagavad Gita, Sri Krishna talks about nature of soul and says we should identify ourselves with the soul and not with the physical body. So the question should be who is owner of the body rather than soul.
The soul is eternal. There is no birth to it. (Emphasis mine)
na jāyate mriyate vā kadācin
nāyaḿ bhūtvā bhavitā vā na bhūyaḥ
ajo nityaḥ śāśvato ‘yaḿ purāṇo
na hanyate hanyamāne śarīre 2.20
For the soul there is neither birth nor death at any time. He has not come into being, does not come into being, and will not come into being. He is unborn, eternal, ever-existing and primeval. He is not slain when the body is slain.
Now regarding who is the owner of soul (you), there are various interpretations. According to Advaita, you are no different from the Brahman. It's due to maya we do not know our real nature. So no room for ownership.
According to Vishishtadvaita, Lord Vishnu is owner of the soul (atma). Pillai Lokacharya in his work Navavidha Sambandham gives nine relations between Brahman and Jiva. Owner - owned is also a relationship. In this context, atma means Jivatma.