First off, be mindful that the Holy See and the Vatican are not exactly the same thing. In short, the Holy See is a non-State polity (another such polity are the Knights of Malta) that leads the Catholic Church, while Vatican City is the actual State. The article you linked to and the EU page it's citing at length are about the EU's relationship with the Holy See. This small subtlety matters because it is the Holy See, rather than the Vatican, that usually engages in diplomatic relationships, treaties, etc.
In short the Vatican isn't part of the EU. Nor is it part of the Customs Union, the EEA, the Schengen Area, the Single Market, the VAT area, etc. It has special arrangements with Italy that date back to the Lateran treaty passed between Italy and the Holy See in 1929. As a result of the latter, there are no border controls between Italy and the Vatican. Even more than other European micro states who are landlocked, it is too small to matter economically. The only treaty or arrangement with the EU that I could locate is the one whereby the Holy See is allowed to print Euros for use in the Vatican.
All of which is to say: the Vatican isn't transposing EU Directives.
What the "meet EU standards" in the BBC article is referring to is something else. It is about the Vatican's notoriously opaque bank, which makes the Vatican a de facto tax haven and (until recent years at least) its bank a mafia favorite for money laundering operations. This further ties into the Holy See being an observer State in the Council of Europe (h/t Uri Granta) and having decided to participate in its anti money-laundering program.