How do you duplicate a smart object in Photoshop that doesn't affect the original?

The original smart object layer needs to be an independent smart object layer.  That is the object must be self contained.  It can not be a placed Link file,  It can be a Placed embedded image or a smart object created from Photoshop layers.   You can create additional independent from independent smart object  using menu  Layer>Smart Objects>New Smart Object via Copy. 

If you use Ctrl+J or duplicate layer the new smart object layer will not be independent it will share the original  smart object layer's Object.  All smart object layers have a associated layer transform.  So it is very easy to create a picture package duplicating a smart object layer a few times then edits each layer's associated layer transform.  So  the layers  content is transform to a different canvas location.  You only need to change to change one object to print multiple duplicated images of various sizes. Mockup templates often user duplicated smart object layers to make updating mockups simple.

Hi there,

Welcome to the Adobe Community!

Could you try right clicking on the blue vector object layer and then choose Duplicate layer, change the Document to New and click OK?

The blue vector object layer will be duplicated onto a new document, you can then import that into illustrator and edit it into a green vector object, then bring it back into Photoshop.

Once it is back into Photoshop then you can right click on the Green vector object layer and then choose Duplicate layer, change the Document to the one that contains the Blue vector object and click OK.

Let us know how it goes.
Regards,
Sahil