Yes, the sushi kind of pickled ginger. (I made mine with apple cider vinegar and sugar and a little salt)
Back on the question about keeping the opened jar - I'm sure I've never gotten sick from eating something pickled that is still in the jar the manufacturer produced it in, assuming I kept it in the refrigerator.
On the other hand, there's always a first time, eh?
Most bacteria or fungi aren't all that interested in something so acidic - but I don't remember if there are exceptions.
On jars themselves, I like them for storage in the freezer. The thing to remember is not to fill it full to the very top with liquid, as that could make the jar burst when it gets frozen.