I don't think they're talking about putting the person's details in the blockchain. That would be illegal in the EU anyway.
The VASP stuff is most obvious for example where you send money from Coinbase to Gemini. They have to pass on your private information, which requires them both to fall under privacy regulation as well as KYC regulation. This is basically the same requirement as for Barclays sending your money to Santander (I believe the threshold is $3000 in USA). VASPs and other obliged entities currently include financial services institutions, estate agents, gambling providers, and "other persons trading in goods over EUR 10,000".
There are similar regulations in many countries about, for example, buying or selling a car for︀ cash. The car dealer has to take your private information and hold it in case︁ the government want it. Should an ecommerce store or service provider expect the reporting thresholds︂ to tighten? I Think so. If you are selling any product or service for over︃ 1000 USD/EUR then you can expect an "obligation to obtain, hold, and transmit required originator︄ and beneficiary information, immediately and securely" before long.
Will they go further? Some governments hate︅ physical cash and want to end it. India tried famously (and badly) to remove high︆ denomination bank notes but Scandinavia have been demonetising successfully. My guess is that they'll not︇ go as far as banning private wallets, but that VASPs and other obliged entities won't︈ be allowed to transact with them.