The correct option for running a serverless data store service on two Docker containers that share resources is to put the two containers into a single task definition using a Fargate Launch Type. This is because Amazon Elastic Container Service (Amazon ECS) offers a scalable and efficient container management service, enabling the execution, cessation, and management of Docker containers within a cluster. By opting for the Fargate launch type, you can deploy your services or tasks on a serverless infrastructure that is overseen by Amazon ECS, which eliminates the need for managing the underlying infrastructure. This approach is particularly suitable for the e-commerce company's development team as it allows for the sharing of resources between the two containers while benefiting from the ease of use and scalability provided by the serverless architecture.