For no particularly good reason, I was under the impression Postgres Aurora was just a Postgres compatibility interface layer on top of a common substrate shared with the MySQL Aurora implementation.
During the same timeframe, many other companies have seen the same risk of AWS taking their open source products and providing competing managed offerings and have chosen various methods of protecting themselves from AWS competition. Many of them (including MongoDB, MariaDB, Confluent, CockroachDB, Sentry.io, Apollo, Graylog, Couchbase) have adopted licenses such as the SSPL, BSL, and even the Elastic license that prevent or strongly discourage AWS from using their products in a managed offering. Others such as Grafana Labs have partnered with AWS (under undisclosed terms) for a managed offering. It's an ongoing tension between companies that want to offer an open source product with a managed offering that monetizes it.
The goal is to remain 100% Postgres compatible. So the network protocol, SQL queries, and tools, all can be used without any change. There are (were?) some (read: very few) caveats/restrictions, though, because of the heritage of, and need to be compatible with, Amazon RDS Postgres.
Disclosure: Member of the founding-team of Aurora Postgres; left the team quite a while back, so my knowledge is quite outdated.
Edit: There's a lot of literature, and public information (articles, conference talks, etc.) published by the Aurora team over the years, which all allude to these facts, and you can use that material to build your confidence that it is in fact Postgres compatible.