I think they're being overly cautious, though, small transactions up to 150 EUR/month should be excluded, and there are ways of avoiding even that for businesses like Bitpay in some circumstances, unless the Netherlands, where Bitpay's EU HQ are located, has decided to implement the directive more stringently.
BitPay have always gone far beyond what regulations or laws have required them to do in terms of KYC and privacy invasion. There are a bunch of other gift card providers on the thread I linked above which serve EU customers, and I've not heard anything about new KYC requirements for any of them. Similarly, there are a bunch of alternative payment processors to BitPay, and again, I've not heard anything about new KYC requirements for any of them.
I'm not a European customer so I cannot confirm though. Perhaps someone who is and has made some purchases recently through non-BitPay means could say if there was any difference or any hints of KYC?
At the end of the day, KYC for any and every payment, no matter how small (getting KYCed to buy a pizza!) is utterly ridiculous. We should not be OK with this.