Totally wish we could upgrade but for some reason AWS have still not implemented any upgrade path for Aurora PostgreSQL other than dump and reimport despite apparently working on it for a year...
With Aurora, the storage layer is swapped out entirely for a distributed storage engine, that I believe is based upon DynamoDB.
The wire protocol and server interface are much the same as regular Postgres, though there are some additional benefits as well as caveats as you might expect
https://docs.aws.amazon.com/AmazonRDS/latest/AuroraUserGuide...
This has really put us off using other AWS managed products and was a major factor in us deciding against using Amazon Elasticsearch Service.