Update: Later June 14
Ars Technica's Glyn Moody writes this "only" applies to providers like Facebook and YouTube and Twitter who sequence content for users. This would not seem to apply to hosting providers. The link is here. The writer's own blog posting on Blogger is here.
EFF tells me the rules would apply worldwide to any content if the company had assets in an EU country. But it's unclear if the company could break off EU assets into separate companies, or the law would apply at the holding company level.
Axel Voss, architect of Article 13, seems hostile to protecting user generated content if it costs publishers anything, since platforms can leverage it so much relative to "legitimate" content. He seems to want to go after fake news, too; link.
This is obviously not very well thought out by anyone.
Major Update: June 20
A committee in Brussels passed both Articles 11 (link tax) and 13 (filtering) today, but a final vote in the EU Parliament probably won't happen until December, Verge story.