Microsoft has launched the ability to have multiple NICs per Azure VM sometime ago but in order for you to have that ability enabled on your Subscription, you need to run the below code on your subscription via PowerShell.
First of all you need to login of course using
Login-AzureRmAccount and then select your subscription if you have multiple and then run the first two commands and the third to check whether the features are registered or not, it will show registering until it's done and then the status will change to Registered.
Register-AzureRmProviderFeature -FeatureName AllowMultipleIpConfigurationsPerNic -ProviderNamespace Microsoft.Network
Register-AzureRmProviderFeature -FeatureName AllowLoadBalancingonSecondaryIpconfigs -ProviderNamespace Microsoft.Network
Register-AzureRmResourceProvider -ProviderNamespace Microsoft.Network