Is it possible to copy a layer from Mapbox Studio Classic and use it in TileMill? Or at least how can I open the layer in say, a text editor and see its content so I can replicate in TileMill. This is the only way I can see that I can get the mapbox.mapbox-terrain-v2 in TileMill and export it as an MBTile.
As others have pointed out,
mapbox.mapbox-terrain-v2 is a vector tile source and is incompatible with Tilemill. Mapbox maps are rendered on the server from sources and styles, after which the resulting raster tiles are sent to your browser. This means that, unless you're uploading or replacing source data, you're never actually interacting with vector tile data stored on Mapbox's servers.
Aside from technical limitations, it is actually against Mapbox's Terms of Service to cache and redistribute hosted maps, which includes
If you would like to use TileMill to generate your own raster mbtiles, you can try using raw OpenStreetMap data and an open source style, such as OSM Bright. See this Tilemill guide for one possible approach.
mapbox.mapbox-terrain-v2 is a vector tile source which cannot simply be adapted for use in Tilemill. Vector tile technology was introduced with Mapbox Studio (alias Tilemill 2, now Mapbox Studio Classic …).
However this guide might help you achieve similar results in Tilemill.