Characters' identities are very much tied to their families in Romeo and Juliet. The feud between the Capulets and the Montagues is at the center of the tragedy, and each character identifies him or herself in terms of which family he or she sides with or belongs to. This is what makes the romance between the title characters so difficult—even impossible. Their individual identities (as well as their individual desires to be with one another as a consequence of their identities) are superseded by the family name and loyalty to the clan.
The inability of Romeo and Juliet to move past their family-defined identities results in the tragedy of the play. Early on, when they meet and instantly fall in love, Juliet is aware that her suitor's family name prevents them from being together. In her famous speech, she asks "Wherefore art thou Romeo?" She ponders why he has to be who he is in terms of his family associations. She wants him to "Deny thy father and refuse thy name." This simply means that she realizes she cannot be with him if he is a Montague. He would have to be a person separate from his family, and in this world, it is not possible for him to be so.
The impossibility of their relationship leads them to concoct their secret plans, which eventually end tragically due to miscommunications and misunderstandings. After their deaths, the families have a road to reconciliation to put their feud behind them. However, while Romeo and Juliet are living, it does not seem possible for them to exist on their own outside of their family connections—thus their romance is doomed.