Hello @Christian Herzog,
I encountered the same error when a Shared Private Link was added.
To resolve this, navigate to the Networking section of Azure AI Search and locate the Shared Private Link.
Delete the Shared Private Link, then retry deleting the Azure AI Search service.
To delete a Shared Private Link (SPL) resource in Azure AI Search using the Azure CLI, run the following command:
az search shared-private-link-resource delete --name <shared-private-link-name> --resource-group <resource-group-name> --service-name <search-service-name> --yes
Please do not forget to click "Accept the answer” and Yes
wherever the information provided helps you, this can be beneficial to other community members.
If you have any other questions or still running into more issues, let me know in the "comments" and I would be happy to help you.