How can I take advantage of the enhancement for error 8152 (https://blogs.msdn.microsoft.com/sql_server_team/string-or-binary-data-would-be-truncated-replacing-the-infamous-error-8152/) in Azure SQL Database? Even when logged in as the Server admin, I can't set the trace flag.
You cannot enable traceflags in azure sql database (single database).
You can enable traceflags when using azure sql database managed instances:
Since the azure sql database service is based on the latest stable sql server version
It's (azure sql database) based on the latest stable version of the Microsoft SQL Server database engine.
And in your linked source it is stated that:
...Note that for now, even in SQL Server 2019 CTP 2.0 the same trace flag 460 needs to be enabled. In a future SQL Server 2019 release, message 2628 will replace message 8152 by default....
We should hope to see this more detailed error message enabled by default for the azure sql database service in the future, once that build is deemed stable.