I think that reading the whole soliloquy makes my reading clearer and your reading less plausible. I can maybe see that if you:
Ignored the whole context of the rest of the play
Ignored the lines in the soliloquy which specifically mention the family names
Ignored the fact that there is a convention in English to refer to people by their first names and leave their family name implicit
Then maybe you would come to the conclusion that Juliet has an objection to specifically Romeo’s first name and not the fact that his name more generally links him to his family. But if you don’t ignore those things, it seems clear to me that Juliet is lamenting the fact that the man she loves has a name (‘Romeo Montague’) which links him to a family who she is not allowed to love.
I think that reading the whole soliloquy makes my reading clearer and your reading less plausible. I can maybe see that if you:
Ignored the whole context of the rest of the play
Ignored the lines in the soliloquy which specifically mention the family names
Ignored the fact that there is a convention in English to refer to people by their first names and leave their family name implicit
Then maybe you would come to the conclusion that Juliet has an objection to specifically Romeo’s first name and not the fact that his name more generally links him to his family. But if you don’t ignore those things, it seems clear to me that Juliet is lamenting the fact that the man she loves has a name (‘Romeo Montague’) which links him to a family who she is not allowed to love.