You're correct that the black finish is more like a paint than dye-stained wood. What it is, exactly, is proprietary, but I suspect it's a pigmented lacquer.
I rather like the way these look as the finish wears and exposes the underlying ash veneer. I wouldn't experiment on them. It's too easy to make things much worse.
You can easily touch up wear on edges with a marker but don't try that on the seat, as you'll never match the color or the sheen-- the whole seat has to be sprayed to be make it even. You might put a simple hard wax on it for some protection and luster but I wouldn't sweat it too much. No india ink, yikes!
If you need any help, please contact us at – info@designaddict.com