There is a connection string on the Azure Management portal's Configure page for the website. This connection string is used to connect to the website's database. It initially contains the User Id and Password used to create the database. Entering a different User ID and Password in the Visual Studio website project (in Web. config or in the Publish Web dialog) has no effect on the connection string in Azure. This caused a problem in my specific situation, but would also be an issue for anyone who does not use the same-like server-level user for accessing the database from the website.
The workaround I found for this issue is to directly modify the Connection String in Azure on the website's Configure page, instead of relying on the effectiveness of a change in the Visual Studio project. This resolved the problem and now the named schema is working correctly.
Regarding Azure SQL databases, SQL Search is primarily designed to work with on-premises SQL Server instances. However, it's important to note that the compatibility of tools and extensions can change over time as new versions are released, and developers create updates.
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