I believe saving SQL select output data in S3 ONLY works for Amazon Aurora MySQL DB. I don't see any reference in the official documentation that mentions the same for Amazon Aurora PostgresSQL.
Here are snippets from official documentation that I referred to
Integrating Amazon Aurora MySQL with Other AWS Services
Amazon Aurora MySQL integrates with other AWS services so that you can
extend your Aurora MySQL DB cluster to use additional capabilities in
the AWS Cloud. Your Aurora MySQL DB cluster can use AWS services to do
the following:
Synchronously or asynchronously invoke an AWS Lambda function using
the native functions lambda_sync or lambda_async. For more
information, see Invoking a Lambda Function with an Aurora MySQL
Native Function.
Load data from text or XML files stored in an Amazon Simple Storage
Service (Amazon S3) bucket into your DB cluster using the LOAD DATA
FROM S3 or LOAD XML FROM S3 command. For more information, see Loading
Data into an Amazon Aurora MySQL DB Cluster from Text Files in an
Amazon S3 Bucket.
Save data to text files stored in an Amazon S3 bucket from your DB
cluster using the SELECT INTO OUTFILE S3 command. For more
information, see Saving Data from an Amazon Aurora MySQL DB Cluster
into Text Files in an Amazon S3 Bucket.
Automatically add or remove Aurora Replicas with Application Auto
Scaling. For more information, see Using Amazon Aurora Auto Scaling
with Aurora Replicas.
Integrating Amazon Aurora PostgreSQL with Other AWS Services
Amazon Aurora integrates with other AWS services so that you can
extend your Aurora PostgreSQL DB cluster to use additional
capabilities in the AWS Cloud. Your Aurora PostgreSQL DB cluster can
use AWS services to do the following:
Quickly collect, view, and assess performance for your Aurora
PostgreSQL DB instances with Amazon RDS Performance Insights.
Performance Insights expands on existing Amazon RDS monitoring
features to illustrate your database's performance and help you
analyze any issues that affect it. With the Performance Insights
dashboard, you can visualize the database load and filter the load by
waits, SQL statements, hosts, or users.
For more information about Performance Insights, see Using Amazon RDS
Performance Insights.
Automatically add or remove Aurora Replicas with Aurora Auto Scaling.
For more information, see Using Amazon Aurora Auto Scaling with Aurora
Replicas.
Configure your Aurora PostgreSQL DB cluster to publish log data to
Amazon CloudWatch Logs. CloudWatch Logs provide highly durable storage
for your log records. With CloudWatch Logs, you can perform real-time
analysis of the log data, and use CloudWatch to create alarms and view
metrics. For more information, see Publishing Aurora PostgreSQL Logs
to Amazon CloudWatch Logs.
Ther is no mention of saving data to S3 for PostgresSQL