Jet indeed derives from wood and is a sort of coal, half way in the process between brown and black coal. Whitby in Yorkshire was a centre for mining jet, hence the term "Whitby Jet" for English jet.
Jet was often imitated by using glass or mixing pulverised jet with the ingredients of glass in the making process. Sometimes the goal was to achieve more luster than jet could offer, particularly in the production in France in the early 20th century and in Bohemia, probably influenced by their glass production. Our idea of "jet black" probably derives more from that sort of jet.
Jet - unlike glass - will remain rather warm if you put it into the fridge, that's one way to identify it. Another is to heat a needle until red hot. You'll be able to bore a small hole with it into jet, but obviously not into glass. And jet, like any coal, will leave a light brown trace on certain materials.
The whole concept of mourning jewellery is very interesting and I only wished I had more time to participate in the discussion, but at the moment I shouldn't even read the board... Thanks nevertheless for bringing up that topic!