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Disclosure: Member of the founding-team of Aurora Postgres; left the team quite a while back, so my knowledge is quite outdated.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.