"Though art thyself, though not a Montague"

Juliet is asking Romeo to "refuse [his] name." He's a Montague, her family's mortal enemies, and as a result, their romance is doomed. She wants him to not be a Montague because she loves him. She promises to " no longer be a Capulet" in return. She wants them to banish their names so they can be together. In the second part of her monologue, she talks about how odd it is that so much is riding on a last name. Your last name isn't who you are as a person. Juliet says that a name isn't even a tangible thing, but yet it stands between them and their love.