Apparently, if you authorise someone else to collect it, you do need to bring proof of travel- so that makes sense. My partner and I can go together and collect each others.
"Those who cannot come in person can authorise other persons to obtain the updated paper vaccination records for them. The authorised person shall present his or her own proof of identity, a completed authorisation form (which can be downloaded from the website of the COVID-19 Vaccination Programme) and copies of the recipient's proof of identity and vaccination record."