Even though an Azure SQL Database stores all data on the Azure cloud, it does not mean that your options for managing and monitoring the databases are limited compared to on-prem databases.  In fact, the options available are very similar to on-prem, including 3rd-party products that support Azure SQL databases.  Here are the various options:

• Overall Azure service health: See Track service health
• Azure Portal: See Managing Azure SQL Databases using the Azure portal
• Alerts: See Receive alert notifications
• Data Management Views (DMVs): See Monitoring Azure SQL Database using dynamic management views
• SQL Database Auditing: See Get started with SQL database auditing and Monitor your Azure SQL Database Auditing activity with Power BI
• SQL Database Threat Detection: See Get started with SQL Database Threat Detection
• Extended Events: See Public preview: Extended Events for Azure SQL Database
• System Center: See Windows Azure SQL Database Management Pack for System Center 2012
• Cotega: See Azure SQL Database Monitoring
• Cloudmonix: See Microsoft SQL Azure Database Management
• Quest Foglight for SQL Server: See Foglight for SQL Server (note the free Foglight for SQL Azure® add-on cartridge)
• Quest Spotlight for SQL Server: See Spotlight on SQL Server Enterprise
• Redgate does not yet support Azure SQL Database with SQL Monitor, but some other products do: SQL Compare, Data Compare, Data Generator, SQL Test, SQL Prompt, and SQL Source Control (minus migrations)
• Idera has SQL Diagnostic Manager, which works with Azure SQL Database but is not officially supported (it will be early this year).  They also have Azure SQL Database Backup (free tool)
• SQL Sentry’s Performance Advisor works with Azure SQL Database