If you saw old fabric wrapped wires, then your original house wiring is quite old. Go back in time far enough and there simply may not be a ground available at the box. If the wrapped wires were in a metallic sheath, the sheath itself was used as the ground. Have you verified no ground with an outlet tester? A GFCI outlet at this location will provide acceptable protection to this and downstream outlets but must be installed and labelled correctly:

https://dfarq.homeip.net/gfci-without-ground-wire/#:~:text=A%20GFCI%20without%20ground%20wire,ground%20wire%20and%20it's%20legal.&text=If%20the%20outlet%20shorts%20out,for%20safety%2C%20even%20without%20ground.

That box reminds me of a surface-mount raceway box; LeGrand Wiremold is one such family, there are others as well. One reason to use such a box is to gain volume; the original DIY'er may not have been comfortable making the new Romex tie-in in the cramped confines of the original box.

If there is no box in the wall behind this surface-mounted box, it probably means some DIYer was too lazy to install a box in the wall. If you want this recessed you're going to need to install a properly sized box that is adequate for whatever wires are coming together there. Since you have a mix of ungrounded old wire, and Romex cable which has a ground wire, you might want to have an electrician look at this to make sure you don't have any improper runs, and that everything that should be grounded is grounded. From your description, it's not possible to determine which wires bring power to the box, and which wires go elsewhere. You could have Romex attached to ungrounded wires running to outlets elsewhere you think are grounded that aren't.