I am looking for a way to “ionic start” a project and having it in the projects folder of an existing Angular CLI solution (containing a web app and a lib already).
So there isn't an official ionic cli command that can do this, but you can do
ng add @ionic/angular
In the Project and it will add the necessary pieces to the web app for you.
Other than that, there's always the manual way of installing ionic/angular
npm install @ionic/angular
npm install @ionic/angular-toolkit