Yep that is because the model you are using is one of our “showcase” pre-made models that are actually wrapped in a nice easy to use JavaScript class the TensorFlow.js team wrote which makes it simple for JS devs to use - no need to worry about Tensors and such to convert webcam image to correct input format for model etc.
The link above applies when working with the raw TensorFlow.js model itself - that is when you are loading the model.json file yourself and loading that directly (which our class does for you behind the scenes) so that class would have to support expose calling that save method which to the best of my knowledge I do not think it does in its current form as these are designed to be run like you currently have it running - cached by browser running as a singleton in the web app.