@Kallin Carolus Khan Thank you for reaching out.
Logical replication is not supported in Azure Cosmos db for PostgreSQL at this time. This is a limitation in Azure Cosmos db for PostgreSQL. It is in our Product group backlog however there is no commitment timeline.
Azure PostgreSQL flexible server does support logical replication. see here https://learn.microsoft.com/en-us/azure/postgresql/flexible-server/concepts-logical
You can raise a feature request for azure cosmos db for PostgreSQL.
Some customers have done this, and it is under review. You can comment and vote on the below post once you access the portal.
https://feedback.azure.com/d365community/search/?q=azure+cosmos+db+for+postgresql
Regards,
Oury