What does it mean when powdered ingredients get thin strings?
I think you are right to figure it is some kind of insect. Little thin threads of cocoonish material in containers of stored food are a pretty clear sign. The strings make themselves noticeable by catching the dust particles from the food. Although it might seem strange to find insect activity in instant coffee, there are tons of different kinds of insects, each one with its own set of tastes. I have found little tiny crawly beige bugs in baking soda even, really. 

When this happens, you need to go through everything, discard every infestation you find, and completely re-do your pantry storage system. Lots of advice available on this site and elsewhere on how to rid yourself of these pests.E.g.: 
https://cooking.stackexchange.com/search?q=How+to+get+rid+of+weevils%3F  , 
and many others. Even though your bugs may not match the exact insect species addressed (maybe they aren't really weevils for example), bugs are bugs, and you don't need a precise id to get some good advice on extermination.