Azure DevOps release pipeline: Angular and .NET Core application

My recommendation would be to implement the following pattern for your pipeline:

1. 'ng build --prod' the angular app in it's own job, and add the artifacts to your pipeline
2. 'dotnet publish' the dotnet core api in it's own job, running in parallel with the angular job, and add the artifacts to your pipeline
3. Append the Angular and Dotnet Core artifacts together into a new artifact. This serves as your final package to deploy
4. Deploy the final package

You're missing step 3, so you'd want something like the following logic defined in YAML, where you create a new zip that represents your actual deployed bits in your pipeline. Then release that artifact, since it is the representation of what you have running on your instances.

  - job: CreateReleaseArtifact
    displayName: 'Package for shared-hosting of angular app and web api'
    pool:
      vmImage: windows-2019
    dependsOn: 
      - BuildNetcore
      - BuildAngularApp
    condition: succeeded()
    steps:
    - checkout: none
    - download: current
    - task: CopyFiles@2
      displayName: 'Copy WebApi Files'
      inputs:
        SourceFolder: $(Pipeline.Workspace)/api
        Contents: '**/*'
        TargetFolder: $(Pipeline.Workspace)/package
        includeRootFolder: false
    - task: CopyFiles@2
      displayName: 'Copy Angular Files'
      inputs:
        SourceFolder: $(Pipeline.Workspace)/webapp
        Contents: 'wwwroot/**'
        TargetFolder: $(Pipeline.Workspace)/package
        includeRootFolder: true
        OverWrite: true
    - publish: $(Pipeline.Workspace)/package
      artifact: package

Does anyone know of a way to ensure the deployments in a given stage simply appends the changes rather than replacing them?

Based on my experience, in your case, after deploy the API or Angular 7, then I you could use the Kudu zip API to upload another one to the Azure WebApp.

You could use the Powershell task to do that. For more inforamtion about powershell demo code, you could refer to this link.

If creating another WebApp is acceptable, you could add a new WebApp and use the same service plan (no extral cost). Then you could deploy them separately.