Data's skin being non-organic, doesn't 'self repair'. If they just sewed or glued on new pieces as his skin wore through at the joints and normal abrasion spots or was damaged in combat or accidents, he would quickly look like Frankenstein's monster yet he has no scars or notable blemishes.
Is it ever shown or explained how his skin gets repaired or replaced?
Your question appears to be based on some incorrect premises. Many things can be repaired without looking like a stitched or glued together patchwork of parts. Transporter/replicator technology alone would allow his skin to be repaired fairly easily without leaving visible scars.
Secondly, Dr. Soong took great pains to make Data human-like down to the smallest details. This is why he's able to eat and drink normal food, breathes, blinks (based on a Fourier series), is anatomically correct, has an aging program, has hair that grows (at will), and is even able to dream. It's therefore likely that Data's skin also self-repairs to some degree like a human's, and there's no evidence that it doesn't.
There are no canonical sources describing Data's skin, but a close-up of his hand shows that his skin (Data refers to it as molecular fabric; in a later episode, it's described as bioplast sheeting) has a cell-like structure:
Lastly, there's no rule that says only carbon-based materials can be self-repairing. Here's a clear counter-example.
I would assume that Data's skin is probably resilient enough to resist fatigue at his joints, or the bioplast sheeting has some basic self-repair functions to counter deformation stresses. When his skin does need manual repair, it would be a simple matter of grafting new bioplast sheeting over the damaged area. If a dermal regenerator can heal wounds without leaving scars, then how Data's skin can be repaired without scarring should require no further explanation.
Here is an image of Doctor Crusher performing surgery on Data (and possibly Lore) from the episode Datalore (S1-E12). She is preparing to close up Data's skin after studying his internal makeup.
This device, whatever it is, does not show up on any of the sites as a dermal regenerator, even though it acts very much like one sealing the skin without a mark.
It could be a prop oversight, where the tool IS a dermal regenerator and simply never used again or...
It could be an engineering tool used to close the bioplastic interface which acts as Data's skin.
In either case, it adds to the idea, whatever Data's skin is made of, it is treated physically as well as medically as an analog similar to human skin, similar enough to be treated medically with the same skills used to treat human beings.
