Relatively sure this would be fine in some European countries. In Germany, afaik it's an ongoing question on whether there's a copyright on the records (especially from the lawyer texts).
The names and so on are always censored anyhow, in some cases it's a bit obvious who did things though. If it's a lawsuit about a company and let's say it sells books online, maybe you can tell by 1-2 things what company it is. But for people, it's not so straight forward.
This is actually starting to become a problem because computers are getting too good at their job.
Let's say a news site reports on a criminal trial of a John Smith, censored as John S. If John Smith was in any way famous before the trial and had an article written about him, that article is somewhat likely to appear in the "you may also like" sidebar when you're reading the censored one. Some news sites try to suppress this, but I'm not sure they're legally required to do so.
I presume since it’s free and is likely domiciled/hosted in America, the owner could probably just scrape euro records and tell the EU “lol get bent” if they made a stink, but probably doesn’t want to run the risk.
Also would be a very strange to apply this so-called “right” when court records are essential to keeping organs of government accountable.