@Suyash Dwivedi Thanks for posting your question in Microsoft Q&A, apologize for any inconvenience caused on this.
Based on the shared information, I understand that you are trying to create a DataSource & index using the built-in serverless sql pool which is getting failed with internal server error.
If my understanding is correct, then this is an expected behavior.
All the built-in SQL serverless pool that is created in synapse won't be shown under your subscription and that is reason you are not able to create a data source with Azure AI search service.
Also, the scope of this serverless pool will be within the workspace only.
We have observed that we can create the data source in AI search with dedicated SQL pool in synapse workspace.
If you want to create a data source with Serverless SQL I would suggest you create an Azure SQL database in serverless mode and try connecting to search service from it.
Refer to the similar thread here : https://learn.microsoft.com/en-us/answers/questions/2088971/unable-to-connect-with-azure-sql-database-through
Hope this helps, let me know if you have any further questions on this.