Yes in this song she is juliet, and she's referring herself in third person, saying her name because her dad told romeo to stay away from her. And shes quoting what romeo said to her, "marry me, Juliet"
And it switches from 21st century to like the 18th century because her love story is like romeo and juliet, her parents don't want her to date him and that stuff, she met him and didn't know who he was but she fell in lvoe with him, etc. So they're acting out romeo and juliet and it flashes to the 21st century to get you thinking oh okay, that's how they are in real life, meet each other at school.
Get it now?