In response to your idea: If you plan to make a lot of this product without variation it may be more economical to use a more traditional manufacturing method.
If you don't plan to make too many items, or want to customize them, or just want to test it out with a prototype, there are many ways you can get this done.
FDM/FFF - Plastics (ABS, PLA, PVA)
SLA - Photopolymer resins
Laminate - Paper
Inkjet - Plasters, Photopolymer resins
SLS/SLM - Huge range of plastics and metals
I've seen inkjet printers print materials with a "rubber" consistency, so its definitely doable.
3D Printing is used for both prototyping and small run custom jobs (aerospace and high end auto are good examples of this). An example of "mass production" 3D Printing would be something like Invisalign teeth braces. It's not mass production in the traditional sense as each brace is custom fit for a single person, but they create thousands of these in a factory setting.