All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
First of all, both of the boy and the girl whom you mentioned in the question are obliged to sincerely repent to Allaah for indulging in fornication, and they should immediately cut the relationship between them. Zina (fornication and adultery) is one of the grave major sins and it is forbidden by absolute evidence. Allaah Says (what means): {And do not approach unlawful sexual intercourse. Indeed, it is ever an immorality and is evil as a way.}[Quran 17:32]. Furthermore, the Prophet said: “When a fornicator (or adulterer) fornicates (or commits adultery), he is not a believer.” [Al-Bukhari and Muslim]
As regards the question whether or not they are considered as a husband and wife, then the answer is 'no', rather they are fornicators who committed one of the grave major sins.
There are conditions for a correct marriage; these are discussed in Fatwa 83629.
Concerning the question whether one of them is bound to the other, then the answer is 'no', rather it is not acceptable for a fornicator, whether male or female, to marry unless they repent to Allaah, and the waiting period of the female fornicator expires. It is not permissible for the male fornicator to marry the woman he fornicated with or any other woman unless he repents. Similarly, it is not permissible for the female fornicator to marry the man who fornicated with her or any other man unless she repents and her waiting period expires. For more benefit, please refer to Fataawa 81160 and 81570.
The fornicators should know that the door of repentance is open and that Allaah Loves those who repent, Accepts their repentance and Wipes out their sins. Allaah Says (what means): {And it is He who accepts repentance from His servants and pardons misdeeds, and He knows what you do.}[Quran 42:25]. Allaah also Says (what means): {Indeed, Allaah loves those who are constantly repentant and loves those who purify themselves.}[Quran 2:222]. Therefore, they should both repent to Allaah and have glad tiding in the mercy and forgiveness of Allaah, provided they had repented.
Allaah Knows best.