Hi
No it will not require a new name ine case of just moving the resource .
But you may know that all resources doesn't support move .
You can see the full list below :
https://docs.microsoft.com/en-us/azure/azure-resource-manager/management/move-support-resourcesIf the move is not supported you will need to create it directly in the new subscribtion .
When it comes to create resources for the new subscription you need to consider the scope for the resource names.
Global = Unique across All of Azure (Name of Paas resources with public ip endpoints accross all Azure .
Resource Group = Unique within the resource group ( Virtual Networks should have unique name when they shared the same resource group but the same name can be used within another resource group and in your case another subscription )
Resource Attribute = Unique within the parent resource( Subnets need to be unique within a vnet but can have duplicate name when related to different vnet .