<p>I am new to 3D printing. I own jewelry stores and want to 3D print my jewelry packaging for rings, necklaces, and bangles as in the picture below:</p>
<p><a href="https://i.stack.imgur.com/NVwgE.png" rel="nofollow noreferrer" title="Product photo of a jewelry ring box"><img src="https://i.stack.imgur.com/NVwgE.png" alt="Product photo of a jewelry ring box" title="Product photo of a jewelry ring box" /></a></p>
<p>I have two main problems:</p>
<ol>
<li><p>Is 3D printing capable of building this package?</p>
</li>
<li><p>I know I can build boxes for jewelry with the outside being made of plastic. But I want the inside to be like a sponge. Is there a filament or a way to print a filament to make it look like cloth or a sponge?</p>
</li>
<li><p>Are there printers on the market which are able to print several copies without the need to set up each time it finishes a single box?</p>
</li>
</ol>

With the right materials
With the right material, you can get flexible surfaces and prints. Just two random examples:
TPU is a flexible material, which can be used to print something like "Lips" that flex and take the jewelry or even strings that suspend the piece in the center.
Foaming TPU is a variant of normal TPU that expands during printing. This makes it somewhat spongey.
However, those have downsides: they don't make good rigid shells, so you will need two different materials: one hard for the shell, and something flexible for the holder.
Luckily, any direct drive filament printer can work with flexible filaments, and there are some flexible filaments that work with a Bowden setup. Due to dissimilar materials though, you need to either assemble the part or buy a somewhat specialized printer: one with two nozzles. These are available but are way out of hobby-grade pricing.
Also, you will never get the "smooth" silky look of a fabric insert cover, but always a clearly industrial printed surface.