Good day ,
using single SQL server hosted in Azure
We must start with clarify this point. When you speak about "SQL server hosted in Azure", do you mean (1) Azure logical SQL Server or do you mean (2) SQL Server which is install on Azure Virtual machine?
In the first case (using Azure Logical SQL Server), you should know that you do not pay for the logical server. It is not a real server but a logical rapper application, which is used to manage different type of database services (on the same logical server you can have different product like Azure SQL Database which is base on SQL Server, or Azure Data warehouse which is base on Parallel Data Warehouse). In this case the databases are totally separated from the user aspect and in fact they are physically might not even be in the same machine.
Therefore, if the number of the client is low, then you can simple create separate logical server for each one.
If the number of logical server is high then you can simply manage LOGINs and USERs according to your need and provide to each client a separate LOGIN and a USER in the database. This is why we have LOGIN and USERs
In addition in order to manage multiple Azure SQL databases, you can use Azure Jobs which allow you to execute queries on multiple databases and manage them as one even if the databases are not physically on the same physical server behind the scenes. For more information you can check this page.
In the second case (using SQL Server which is install on Virtual Machine), you manage it just like on-premises using LOGIN and USERS
In both cases, you should check the option of using Contained Databases