Q: "Is there some standard system of pronunciation to which this [rendition of the anthem of the European Union] conforms, or are they just pronouncing it wrong, or what?"
I agree with the previous answer and comments which suggest that the particular rendition linked in the question is mostly influenced by German tradition.
Relevant to a EU-wide context, I believe that the shortest answer would be that there is no standard. This is because Latin pronunciations -- especially in music -- developed separate traditions in different European countries, all different and all changing over time, and maybe in some places also more or less influenced to different degrees by attempted modernizations of Church Latin. So it might be quite hard, if not impossible, to find agreement Europe-wide let alone worldwide on any standard.
Probably the most comprehensive book published on traditions of Latin pronunciation in music is "Singing in Latin", by Harold Copeman (1918-2003), (published by the author, in Oxford, 1990, with revised versions in 1992 and 1996, and a pocket edition). It has chapters on 'How Latin pronunciations multiplied', about documentary evidence from several national traditions in Europe, music examples, recommendations for singing Latin music while respecting national styles from different historical periods, and a bibliography partly classified by historical period.
Copies can be hard to obtain now, probably because the author acted as his own publisher. At the time of writing this (Aug 2023), a brief pdf extract of the book (with front matter, table of contents, and an example section) could be found online. 'Singing in Latin' also received an extensive review in Performance Practice Review (1992), 5(1), article 3, pp.103-108, written by the late Douglas Leedy.
There is also a short but useful study of "National Pronunciations of Latin ca.1490-1600" by Ross Duffin, in Journal of Musicology 4(2) (1985-6), 217-226.
Likewise, there is a wikipedia page on Latin regional pronunciation, with a summary table of pronunciation points, but it does not discuss changes and developments that have happened over time.