The link between Carnatic and Hindustani Classical is on same lines as Tamil and say Sanskrit they have same origins . Hindustani classical did not come into existence during persian times, it always existed . The convenient boundaries that we draw in terms of north vs south are modern superficial interpretations.
Adi Shankaracharya had travelled across entire India , I am quoting this fact because the knowledge of Vedas across India was spread mostly by the Shankaracharya and his disciples. That's why even to this day the head priests of Kedarnath and even Pashupatinath temple in Nepal come from Southern India (appointed by Kanchi )
How is this relevant to Indian classical music scene ? The classical Indian music as we know today dates back to the Vedic times (honestly could go far back) , however the current consensus is Samveda is the first well known written record of a treatise on music and Natya shastra. The Carnatic music and Hindustani Classical are fundamentally the same thing . However the Hindustani Classical diversified into Thumri , Khayal and other styles over a period of time due to various influences. The Indian classical music was mostly devotional and meant for conducting ceremonies in temples or key facets of life. However as the north saw a constant assimilation of foreign occupiers, music was re-purposed for entertainment by the ruling elite . A good example is "Khyal"
https://en.wikipedia.org/wiki/Khyal ; if you observe there has been no fundamental change to the Ragas . This phenomenon is true for dance forms too , if you notice Kathak has been repurposed for entertainment as for Mughals these art forms were primary sources of entertainment, Bharat-Natyam, on the other hand, is still much closer to it's original form.
Another key marker is , old Hindustani classical style never made use of the Tabla , instead Pakhwaj was used (it's similar to Mridangam in south). However, during the Mughal period as Khyal and other styles emerged Tabla replaced the Pakhawaj.
Art forms evolve, even Carnatic music today is not pure in the sense that it is not being rendered in the same manner and with the same instruments when it originated; a good example is the usage of the violin. Violin was not part of the original set of instruments even when this art form emerged in the south thousands of years ago.