Because you need a way for the pub to authenticate that this is indeed your qr code (matching your id)

The EU standard was developed for the purposes of avoiding additional quarantine or testing during cross-border travel, not going to pubs within one’s own country. For domestic use of proof of COVID vaccination, some countries developed their own internal standards alongside the EU standard.

Not currently visiting pubs, I still wait until they stop validating their guests.

I don't think name and date of birth are enough for identity theft anywhere in Europe...

To check with your passport/id card that this is your QR code and not someone else’s…

To compare it with name and DOB in the government ID.

I would guess so that they can use more traditional ID (like a drivers license) to confirm the vaccine record is yours.