Hasan Savran explains the scope of unique keys in Cosmos DB:
I wrote about Unique Keys and tried to explain how they work in one of my earlier post. It’s common to use SQL Server’s Primary Key or Unique Indexes to explain Unique Keys of Azure Cosmos DB. If you have a Primary Key in a table in SQL Server, the key you defined cannot be in that table more than once. By adding a unique index or unique constraint in a table, you guarantee that no duplicate values can be in your table. The key word in both of those statement is the TABLE.
Azure CosmosDB Unique Keys do not work like Primary Key or Unique Indexes/Constraints
Read on to learn how Cosmos DB differs.