The code was working after adding a 'reader' role for principle user in sql server.
The steps that are required to connect SQL Server through AAD - Managed identity for NodeJS project are as below:
Create sql-server with sql-server database in an azure resource.
Create an azure-web app within the same azure resource.
Create a principle user in the web app.
This can be done by going to the azure web app > Settings > Identity menu > System-assigned tab and turning on the toggle.

In the sql-db create the user with the same name as principle user that was created in step 3. The name is same as web-app.
Also assign db_reader, db_writer, and ddladmin roles to the user.
You may use the below script to create and assign role to the user. (One thing to consider if it fails to create the user, you might need to rename the app-service name as in my case it was not allowing me to create the User saying it already exists).
**CREATE USER [<identity-name>] FROM EXTERNAL PROVIDER;
ALTER ROLE db_datareader ADD MEMBER [<identity-name>];
ALTER ROLE db_datawriter ADD MEMBER [<identity-name>];
ALTER ROLE db_ddladmin ADD MEMBER [<identity-name>];**
- Add the reader role to the principle user in the sql server. (The one that I was missing).
This can be done by going to the Sql server resource(in Azure)> Access control (IAM)

Select Reader and select member/user (principle user with the app name), and assign the role reader to that user/member.

- After all these configuration part is done, use the code from microsoft docs. For me lower node version was not working so needed to upgrade the node version.(14 in my case).
https://learn.microsoft.com/en-us/azure/azure-sql/database/connect-query-nodejs?tabs=windows
Also the connection was successful only in the deployed version.
Hope this is helpful !! :)