Is an oath and a covenant between fiancés considered a legitimate marriage?

Marriage is not Islamically valid except with its pillars and conditions, such as the formula of offer and acceptance between the husband and the wife's guardian, and the presence of witnesses. Your cousin's statement that you are his wife before Allah does not constitute marriage. Therefore, if the suitor is religious and of good character, accept him as a husband, and hasten to complete the marriage, acting upon Allah Almighty's saying: "So vie with one another in good deeds. Wherever you may be, Allah will bring you all together. Indeed, Allah is over all things competent." (Al-Baqarah:148).