How to create this image

I would like to create an image like the one shown in Picture 0 and 1. This image was created using Adobe Character Animator. When this image is placed on any object, it looks as if the mouth is actually coming out of that object. For reference, please see Pictures 2 and 3. How can I create such an effect using Blender?

You might draw a simple depth map for the lips.

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Using geometry nodes, it can be spread out in the same plane as UV.

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