AWS Systems Manager Session Manager allows you to connect to an instance in a Private Subnet because the instance is actually running an 'SSM Agent'. This piece of code creates an outbound connection to the AWS Systems Manager service.
Then, when you request a connection to the instance, your computer connects to the AWS Systems Manager service, which forwards the request to the agent on the instance. The AWS Systems Manager service is effectively acting as a Bastion for your connection.
AWS Systems Manager Session Manager cannot provide a connection to an Amazon RDS server because there is no ability to 'login' to an Amazon RDS server. Given that your RDS server is running in a Private Subnet, it is therefore necessary to port-forward via an EC2 instance in the same VPC as the RDS server. This can be done via a traditional Bastion EC2 instance in a Public Subnet, or via an EC2 instance in a Private Subnet by taking advantage of the Port Forwarding capabilities of AWS Systems Manager Session Manager.