I have an application that consists of microservices which are dockerized and are deployed on Azure kubernetes service in West Europe.
In order to reach the application an nginx ingress controller is created and the public endpoint is mapped to a custom domain. For example the public ip x.x.x.x is mapped to domain testwebsite.com in Azure DNS. The Ingress takes care of the routing to the microservices.
How do I translate this setup to multiple regions and still use the same DNS name?
An ingress controller is a piece of software that provides a reverse proxy, configurable traffic routing, and TLS termination for Kubernetes services. Kubernetes ingress resources are used to configure the ingress rules and routes for individual Kubernetes services.
So ingress is not appropriate to your purpose. According to your description, I think you could try Azure Traffic Manager. The introduction of Azure Traffic Manager is here:
Azure Traffic Manager is a DNS-based traffic load balancer that enables you to distribute traffic optimally to services across global Azure regions, while providing high availability and responsiveness.
With this introduction, I think it's more appropriate for your purpose.