Good question.
Actually it is way more deeper than we think. In fact, Shri Krishna has said that it is only out of ignorance that people assume that he is in the process of birth and death like other living beings.
See, Almost every deity that we worship in Sanatan Dharma has one unmanifested form (mulswaroop) and one avatar swaroop and multiple other avatars as well. This doesn't reduce the Ishwaratva (Godhood) of the deity. This applies to all the deities, I mean all the forms of the eternal truth.
Shri Krishna is considered as an incarnation of Bhagwan Vishnu himself but he is also considered as the unborn Supreme entity. Then the question arises that how come he is unborn if the scriptures clearly describe him taking birth as the son of Vasudev and Devki?
We must know that Bhagwan Narayan did give them darshan before taking birth as their son. Can an ordinary human being tell his parents that he will take birth as their son? Definitely not!
But still, in order to remove the confusion, Bhagwan Krishna himself has answered this question in Bhagavad Gita. While giving the Updesham to Arjun, Shri Krishna says that he gave the same knowledge to Surya Dev during the time of creation. When Arjun asked this question 👇
अपरं भवतो जन्म परं जन्म विवस्वतः।
कथमेतद्विजानीयां त्वमादौ प्रोक्तवानिति॥
Meaning– You were born in recent time and Surya (sun God) was born during the start of the cosmic cycle, he is very ancient then how should I understand this that you can give this Yoga to him?
Arjun knew Shri Krishna as his relative, someone who was highly intelligent and divine, but he didn't know fully about the identity of Krishna. Then Krishna replied in this way.
बहूनि मे व्यतीतानि जन्मानि तव चार्जुन।
तान्यहं वेद सर्वाणि न त्वं वेत्थ परन्तप॥
Meaning– O Parantapa Arjun, you and I have had many births. You don't know all of them but I know them.
Here Krishna clearly explained Arjuna the difference between Parmatma and Jivatma. While scriptures do talk about the oneness of Brahman and Jiva, they practically function in different way, Parmatma is aware about everything and himself chooses to act (lila) while Jivatma is under the influence of maya and is unaware about the actual reality. So when God incarnates, he knows when, where and how he will take birth and when how will his life end, but on the other hand we are not aware about our birth and death, these things are not in our control.
After this the famous shloka comes.
यदा यदा हि धर्मस्य ग्लानिर्भवति भारत।
अभ्युत्थानमधर्मस्य तदात्मानं सृजाम्यहम्॥
परित्राणाय साधूनां विनाशाय च दुष्कृताम्।
धर्मसंस्थापनार्थाय सम्भवामि युगे युगे॥
(Bhagavad Gita Chapter 4 verse 7)
Whenever there is a decline of righteousness and a rise of unrighteousness, O Arjuna, I manifest myself on earth. To protect the righteous, destroy the wicked, and establish righteousness, I take birth from age to age.
During this epic conversation, Shri Krishna then shows Arjuna his Vishwaroopa to clear his doubt, thereby proving that he is not a man but the Universe itself.
So Shri Krishna is not limited to his human birth in Mathura, in fact him being Narayan himself, he is not restricted to even one abode Vaikuntha or Vishnu Loka. The same Krishna actually lives as Paravasudeva in Goloka with Radharani and simultaneously resides in every anu (अणु) that's why known as Vishnu.
Whenever God takes a form, that form which we think is comparatively new, that is also eternal. Even Shri Ram in his human incarnation, was born in Ayodhya. But he is the same Para Brahman who lives in Saket Dham with Maa Sita (Lakshmi ji). In fact, this also applies to Maa Lakshmi, though she is described as the daughter of ocean since she emerged from it, As Mahalakshmi is eternal and unborn.
In the same way, if we talk about Bhagwan Shiva. He lives in Shivloka as Sadashiva (Mahakailasha) which is beyond material realm.
He also lives in Kashi as Maheshwar and at Kailash in the form of Rudra.
Then as Bhairava also, he is in the role of Kshetrapala, but is also the Ultimate Power as Parabhairava. And obviously, as Parashiva he is even beyond form and is omnipresent, “Kankar Kankar me Shankar”. Satyam Shiv Sundaram.
In the exact way, Adi Shakti Durga lives with Shiva in Shivloka.
As Kali, she is the same omnipresent Cosmic energy.
Then as Rajrajeshwari Lalita Tripurasundari, she lives in Manidvipa (Devi Loka).
As Parashakti, she is The same Maa Bhavani gave Updesham to King Himavan before taking birth as his daughter Parvati. It is mentioned in Devi Gita.
Then comes Shri Ganesha who is often worshipped as the son of Shiva and Parvati. But that is also an avatar of Ganapati because as Mahaganapati, he is also unborn and eternal Brahman only.
One Para Brahman can have as many forms, infinite forms and all of them can be independent and can expand further. So in this way, all these deities have one mulswaroop, even of the avatars there is one eternal form and then the particular avatar that comes to the mortal world. Shri Krishna is also the same Brahman only.
So I hope this would clear the doubt that Shri Krishna is unborn originally, for the purpose of lila and establishing Dharma, he takes the incarnation.
Jai Shri Krishna 🙏