I can understand why this is upsetting for you, but actually it sounds like the feedback was not around your competence to do the work, but your skill at doing interviews. It's a bit like exam results - doing well essentially proves you're good at exams, not good at maths or whatever.
I'm actually reviewing a bid response some of my team have done at the moment and I think they've fallen into a similar trap. The questions call for detailed, specific answers and they've written fluffy, 'sales' speak. Do you think maybe that's what you did too? Understanding the format and expectation of the interview is half the battle. Of course there's always an aspect of selling yourself as part of an interview but it's judging what the right mix is.
Looking back now, can you think how you would answer the questions differently? I think you need to discuss this with the hiring manager when he/she is back from leave.
As this has come as a surprise to you, that's also something to explore. Have you not been interviewed or been an interviewer recently? Or was this style different from what you'd experienced before? In either case, can you look to gain further interview experience, preferably as an interviewer?
I would look on this as an opportunity to demonstrate resilience and ambition to the hiring manager. I would say you're disappointed that you didn't perform well enough to be considered and you're very keen to put in the work to get to the next level. If the manager is half decent, he/she will work with you to create an action plan (possibly as part of your appraisal?) to get you there next time a job crops up.
It sounds like you're also worried about what other people will think - did the team know you had applied? Did they think you were a shoo-in as well? I think you can frame this as having done a bad interview and needing more practice, you don't have to labour it with them. I assume you're now going to have to work for the person who did get the job - I'd maybe clear the air right at the beginning with them, so they know you applied and didn't ace the interview, but you are a professional who will work with them to make the role a success. Maybe worth highlighting as well, though, that you will be looking around for a similar role.
All in all I'd have one good wallow in your disappointment, then suit up and get back in there to get the most out of this opportunity.