When Brahmin Rishi Kashyap and his Brahmin wife's youngest son Vamana(vishnu avatar). Takes the three steps donated to him by king Bali. He takes the first step on pataal second on the heaven, while doing so his toe pierces a hole in the heavens and outflows Ganga.
Hence she is called as Vishnupadi. So her origin or birth happens by a Brahmin Vamana. Hence by birth a Brahmin
When she reaches earth at the request of Bhagirath, after being released from the Jata of Shiva, she is hurling in a spate and destroys the ashram of Brahmin Rishi Jahnu, who in anger then drinks her up. Other Rishis devas Bhagirath then intervene, and Rishi Jahnu then releases her again from his body through his ear.
Hence she is called as Jahnavi, so her adoption on earth or rebirth on earth is through Brahmin Rishi Jahnu. Therefore she by re birth or adoption as well she is a Brahmin.
Santanu does not call her Ganga he calls her Jahnavi (daughter of Jahnu)
People in that era were generally addressed by their, father or mother or by Varna name.
Bhishma was called Gangeya, Karna was called Radheya or sometimes Suta.
Yudhishtir was also called Kounteya, Pandavas etc.
Usually it depended on how famous your parents were.
If you are son of Ganga. Then everyone would address you as her son. Because he is son of the famous Ganga.
Similarly Yudhishtir and his brothers were son of the famous Pandu who had conquered several kingdoms hence mostly addressed as Pandavas.
Have you noticed though Duryodhan is son of Dhritrashtra he is mostly addressed as Bharata or Kourava. Because Dhritrashtra had never done anything to make him famous enough for his sons to be addressed as his.
The priest of Drupad who goes with the first peace message is addressed as son of Brahmans as his father was nobody worthy enough of mention
Same with Karna. If his adopted father was someone famous then he would rather be addressed as son of that famous man instead of son of suta.
As far as poor are concerned, they were treated badly for their poverty not for their varna
Because they were helpless.
You can notice the difference between Duryodhan and Draupadi from this itself.
The way Duryodhan treats his servants by yelling at them, and the way Draupadi would eat every day only after she had fed even handicapped beggars who came to her door