The misunderstanding in this case I think stems from Julia being a first name in modern times, which is functionally pretty close to a Roman praenomen, to distinguish an individual inside a gens or family, even though Roman praenomina had considerably less variation.

Women where commonly identified by their nomen gentile in the feminine form, in this case Iulia, and carried no praenomen. So one would indeed expect her to go by Julia. However, in some cases, as here, women could also bear a cognomen, in most cases this was simply something designating their order of birth (Prima, Secunda, Tertia, Quarta and so on), in her case it was taken from her mother Agrippina maior who in turn had that cognomen from being a daughter of M. Vipsanius Agrippa. By the end of the Republic, the cognomen became more important as an individual identifier, with people being identified mainly by their cognomen (like Cicero or Caesar - or Agrippina minor & maior).

Note: Caia was in use as a praenomen for women, but to my knowledge only as a placeholder simply meaning "woman", used for example in the nomenclature of freed slaves to identify their former mistress (patrona).

Hope this serves as a quick answer to a quick question, if you want to ask more, feel free to do so!