DCC (Digital Covid Certificate it's called, not Digital Green Pass) is essentially a spec for a QR code (as demonstrated nicely in this post) + an EU signing gateway which is used for signing the certificates. The EU acts similar to a CA in case of SSL certificates.
They could also produce a fake vaccination booklet, or fake Covid test result. I'd guess both of these would be easier. All will result in forgery charges if caught. I think the chosen approach is pretty solid for the purpose.