As an application developer who connects to both Postgres DB and Aurora databases, I can confirm Aurora is far slower that a properly setup local Postgres install.
Its dog slow for online transaction processing. To get the best out of it you have to move all your processes close to it. Even then its slower than an in house setup.
Yes, it scales...supposedly, but then I have not really pushed it at scale as its ingestion rate is rather poor.
Also there was a previous puff piece on here claiming in very wormy word terms that Aurora was open source as its Postgres. Well that's a complete lie. The backend is completely proprietary to Amazon.
I don't know where they get their stats. I think they genuinely make them up.
Another team I work with internally is moving off Aurora as they used it from day one for their business and have had no end of pain making it scale. Sort of made 'Postgres' a dirty word for them, but then they have to be reminded they are actually using Aurora.