After scanning the Azure and AWS pricing pages and calculators for comparably speced Azure Manged SQL and Amazon RDS for PostgreSQL instances, AWS's RDS comes out at about 1/3 the price of Azure Managed SQL.
This matches my experience with deployments on AWS and Azure. Both are managed db instances, but Azure's is consistently more expensive for comparable offerings on AWS or GCP.
I do not have experience with AWS RDS so I cannot speak to that.
In my experience, GCP Cloud SQL for Postgres has been more expensive than MS Azure SQL. In our tests, CloudSQL also was not comparable to the resiliency offered by Azure SQL. Things like Automated DR and automated failover etc. were not at par with what Azure offered. Not to mention , Column level encryption is standard for Azure SQL.
It's been a while since I've looked at elastic pools in Azure, but maybe Neon (https://neon.tech/) or recently Nile (https://www.thenile.dev/) might work in terms of spinning up a bunch of separate logical DBs with shared compute/storage.
This matches my experience with deployments on AWS and Azure. Both are managed db instances, but Azure's is consistently more expensive for comparable offerings on AWS or GCP.