It seems like you have come up with a good solution to manage connection strings for multi-tenant Azure applications. Storing connection strings inside Azure Key Vault is a secure way to manage secrets, and using Azure App Configuration to create references to the Key Vault connection string values is a good way to centralize the management of connection strings.
Your approach of configuring the App Services to access the App Configuration and retrieving the connection string based on the Tenant ID provided by the user is a good way to dynamically select the correct database for each tenant.
You are correct that using this solution, you can change the value of the connection string in Key Vault and the App Services will automatically use the updated value without the need to restart the App Service. Similarly, you can add a new connection string to the Key Vault, add a reference to its value to the App Configuration, and the App Services will be able to retrieve this new connection string without the need to restart the App Service itself.
Overall, your solution seems like a good approach to manage connection strings for multi-tenant Azure applications. Let me know if you have any other questions or if there is anything else I can help you with.