The EU's right to be forgotten says that you can ask search engines to remove search results about you. It has nothing to do with government surveillance.
That's actually not true. There isn't even any actual "right to be forgotten", there just are data protection directives and laws and the like, which were the basis for the ECJ decision that got called the "right to be forgotten" which happened to be about search engines, Google in particular. The same laws do apply to the processing of personally identifiable information in any company, and that in principle should include postal service companies in the EU, as well as to the government itself, though sometimes with limitations.
That does not mean that postal services in the EU don't do the same thing for various reasons, but the legality is questionable.
The EU has it out for Google in particular. Why don't they go after the sites that have the information in the first place?
Regardless, if you have a news story about you, that you did something, it should be there as information forever. If the story is correct, then it's there. 20 years from now, you shouldn't have the right to say to some search engine that just happens to index it that you don't want it indexed anymore. The facts are still the same. The news story is still accurate. The site that has the news story still has it. Just because it may paint you in a negative light 20 years ago still doesn't mean you get the right to expunge that from the public record.
The implications of this are just chilling and far reaching.
Noone is "going after" anyone. Some guy sued Google, the court decided the case they were presented with (and said some things that are applicable outside the specific case).
As for the actual matter, I disagree. In particular, there are quite a few steps between "expunge from the public record" and "make it the most visible thing on the web", where the latter quite clearly is in conflict with the goal of resocialization in the justice system, for example.