Does AWS automatically perform database snapshot backups before and after the minor version upgrade for RDS and Aurora engines? Is this process same for all RDS database engines and Aurora?

For an RDS instance, the answer is yes. During the automatic minor version upgrade process (AMVU), RDS takes two database snapshots during the upgrade of both minor. It will also do this for major version upgrades.

It takes the first snapshot before starting the database upgrades. It acts as a restoration point during the rollback, if required. You can restore the database snapshot if you face any issues after the database on a new version. The second snapshot is immediately after the RDS version upgrade completes

Note: You should have set the backup retention period greater than zero to take advantage of this behavior. If not, take a moment to modify the RDS instance by setting the backup retention period to a value greater than zero.

More details are at https://aws.amazon.com/premiumsupport/knowledge-center/rds-required-maintenance/#:~:text=Upgrades%20to%20the%20database%20engine,size%20of%20your%20DB%20instance..

I believe the above answer by @kekline applies to RDS PostgreSQL, But not to Aurora PostgreSQL.

I was performing a minor version upgrade to an Aurora PostgreSQL instance earlier this week, and was surprised to find that there were no automated backups taken before and after the upgrade. I was going from PostgreSQL 12.7 to 12.13.

However, I was able to perform a point in time recovery of the upgraded database to a time just before the upgrade began. The recovered instance was successfully created and showed the earlier PostgreSQL version.

The documentation ( https://docs.aws.amazon.com/AmazonRDS/latest/AuroraUserGuide/USER_UpgradeDBInstance.PostgreSQL.html) also points out "When you initiate the upgrade process to a new major version, Aurora PostgreSQL takes a snapshot of the Aurora DB cluster before it makes any changes to your cluster. This snapshot is created for major version upgrades only, not minor version upgrades. "