If you are referring to EUID (not fully sure as you said EU eID, i dont know if you are referring the estonia of eID like system)

I have to mention that EUID is not private, since there's "provider" element which informs website if you are 18 or not. The flow is:

1) You scan QR code 2) Your EUDI wallet does verification, informs provider to tell you are 18+ 3) Provider informs website you are 18+

The EUID draft doesnt mention tech like ohttp for anonymizing requests. So there's risk of provider keeping track of who you are. So while everybody claims its fully anonymous which is just false. Government could ask website/service for the token or account information then use timestamp or token then combining with "provider" logs, your identity will be exposed.

EUID has another problem which is letting all countries implement system, which is wasteful duplication effort so this probably will be outsourced and to same company to reduce duplication efforts. Then it'll be centralized and they happen be collecting telemetry data for "experience improvements" as everysite out there do.

I haven't even mentioned biggest problems like requiring attestation Apple/Google. While spec doesn't require it, but the likehood country's app requiring it will be very high.