I want to know more about Yami. Is it believed that there was a twin sister of Yama named Yami?
The story is described in chapter 77 of the Markandeya Purana:
The daughter of Vishwakarman [the divine architect] was the wife of the sun, by name Sajna who begat on her a son, by name Manu, who, illustrious as he was, was equally a master of various branches of learning. He was the son of Vivaswat and was therefore called Vaivaswata. As soon as she was espied by the sun Sajna used to close her eyes and therefore he, in anger, addressed to her cruel words: "Since, O stupid one, you close yours eyes as soon as I cast my looks on you therefore will you give birth to Yama, the destroyer of creatures." Thereupon the goddess, stricken with fear, assumed trembling looks, at which the sun again said to her. "Since beholding me, you have assumed trembling looks you will give birth, as your daughter, to the fickle river." Thus by the imprecation of her husband she gave birth to Yama and the great river celebrated under the name of Yamuna. Henceforth she, with great difficulty, began to suffer the effulgence of the sun.
By the way, in case you're interested that isn't how the story ends: after that Sanjana leaves Surya, because she cannot take his brightness and she's replaced with her shadow Chaya, who pretends to be Sanjana. You can find out about the rest of the story in my answer here.
Yes they are twins. Their mother is Sanjna and father is Surya.
A tale explains their name are as:
Sanjna was unable to bear her husband, the sun's heat and light and closed her eyes in his presence. Surya felt insulted and said that her son will be known as Yama ("restraint"), due to the restraint she showed. Thereafter, Sanjna tried her best to keep her eyes open, however she flickered them angering Surya again who proclaimed that her daughter would be Yamuna. Since Sanjna had tried to keep the eyes open, Yamuna was blessed that she would worshipped as a goddess and remembered throughout time. (Source: Wikipedia).
The confusion can be due to their marriage.
The Rig Veda, in the tenth Mandala, contains a hymn in which they sing to each other. Yami proposes marriage to Yama but Yama flatly refuses as he considers it incest. The Atharva Veda also has a variant of the tale. In both texts, Yami argues that since they have slept together inside their mother's womb together, it was not wrong to sleep together outside it. However, the hymns abruptly end, without the final output being known.In later version of the tale, they are depicted as the first human beings and it is conjectured that Yami was successful in becoming Yama’s wife. (Source:Wikipedia)