Did Satyavatī and Śantanu belong to the same caste (birth-based varṇa)?
No. Śantanu was a kṣatriya and Satyavatī a śūdra.
Satyavatī's biological parents
- Father - Uparicara Vasu (a king, most likely a kṣatriya king)
- Mother - Adrikā (a fisher-woman, so most likely an outcaste or śūdra)
Here's the official birth story of Satyavatī from the Mahābhārata:
And the king [Uparicara Vasu] became possessed with desire, and he saw not his wife before him. Maddened by desire he was roaming hither and thither, when he saw a beautiful Asoka decked with dense foliage, its branches covered with flowers. And the king sat at his ease in the shade of that tree. And excited by the fragrance of the season and the charming odours of the flowers around, and excited also by the delicious breeze, the king could not keep his mind away from the thought of the beautiful Girika. And beholding that a swift hawk was resting very near to him, the king, acquainted with the subtle truths of Dharma and Artha, went unto him and said, 'Amiable one, carry thou this seed (semen) for my wife Girika and give it unto her. Her season hath arrived.'
The hawk, swift of speed, took it from the king and rapidly coursed through the air. While thus passing, the hawk was seen by another of his species. Thinking that the first one was carrying meat, the second one flew at him. The two fought with each other in the sky with their beaks. While they were fighting, the seed fell into the waters of the Yamuna. And in those waters dwelt an Apsara of the higher rank, known by the name of Adrika, transformed by a Brahmana's curse into a fish. As soon as Vasu's seed fell into the water from the claws of the hawk, Adrika rapidly approached and swallowed it at once. That fish was, some time after, caught by the fishermen. And it was the tenth month of the fish's having swallowed the seed. From the stomach of that fish came out a male and a female child of human form. The fishermen wondered much, and wending unto king Uparichara (for they were his subjects) told him all. They said, 'O king, these two beings of human shape have been found in the body of a fish!' The male child amongst the two was taken by Uparichara. That child afterwards became the virtuous and truthful monarch Matsya.
After the birth of the twins, the Apsara herself became freed from her curse. For she had been told before by the illustrious one (who had cursed her) that she would, while living in her piscatorial form, give birth to two children of human shape and then would be freed from the curse. Then, according to these words, having given birth to the two children, and been killed by the fishermen, she left her fish-form and assumed her own celestial shape. The Apsara then rose up on the path trodden by the Siddhas, the Rishis and the Charanas.
The fish-smelling daughter of the Apsara in her piscatorial form was then given by the king unto the fishermen, saying, 'Let this one be thy daughter.' That girl was known by the name of Satyavati. And gifted with great beauty and possessed of every virtue, she of agreeable smiles, owing to contact with fishermen, was for some time of the fishy smell. Wishing to serve her (foster) father she plied a boat on the waters of the Yamuna.
Now if you are like me and find a part of the above story (i.e., a fish getting pregnant by simply swallowing human seed and then delivering a pair of human babies) unbelievable, this article offers a more rational explanation:
This was the strangest myths I could find in Mahabharata. It reveals how imaginative these people were and how they entwine ignorance and imagination into such deadly cocktail. At least some of the ancient people believed that babies are born with the sole contribution from men. They considered women as mere vessels to carry the babies of men. It was unknown to them that to develop an embryo contribution from both man and woman are needed. A fish after swallowing semen delivering babies! Below is a plausible reconstruction of what might have happened.
Adrika was a maiden belonging to the Matsya (fishermen) tribe and Vasu beget two children in her, probably when he was away from his wife Girika. Adrika was described as an Apsara, due to her beauty. Apsaras were mentioned as extremely beautiful. The word 'Apsara' can be divided into 'Apa' (water) and Sara (lakes). They were often described as women seen in the vicinity of water-bodies like lakes and rivers. A Matsya lady (fisher-women) too lives close to water-bodies like seas, lakes and rivers. There are also myths about fairies called Matsya-Kanyakas (fish-women / fisher-women) who were half fish and half woman. Such tales were also similar to the tales of Apsaras. Thus Matsya Kanyakas and Apsaras seems to be same. Both these terms later assumed the meaning of fairies living close to water bodies, and Apsaras in particular was formerly their tribal name. Fisher-women were the Apsaras and the Apsaras were the fisher-women.
Adrika probably died while delivering the twin babies. Death of women after delivering babies, especially twins, were quite common in those days. The fishermen after learning about the affair between Adrika and the king might have taken the babies to the palace.
Assuming Satyavatī was born to a Kṣatriya father (Uparicara) and a Śūdra mother (Adrikā), she must be an Ugra1. And because an Ugra doesn't fit into any of the four castes2, she's an outcaste. However, for all practical purposes she can be considered a Śūdra3.
1 From the Kṣatriya on a Śūdra maiden is born a being called ‘Ugra,’ of the stuff of the Kṣatriya and Śūdra, cruel in his deeds and dealings. [Manu 10.9]
2 The Brāhmaṇa, the Kṣatriya and the Vaiśya are the three twice-born castes; the fourth is the one caste, Śūdra; there is no fifth. [Manu 10.4]
3 Six sons born on women of the same caste and of those of the next lower castes partake of the character of ‘twice-born’ persons. But all those born of violation have been declared to be of the nature of Śūdras. [Manu 10.41]
From here, Śantanu appears to be a pure Kṣatriya (i.e., no intermixing of castes):
And Pratipa married Sunanda, the daughter of Sivi, and begat upon her three sons, viz., Devapi, Santanu and Valhika. And Devapi, while still a boy, entered the woods as a hermit. And Santanu became king.