It is convergent evolution.
You are thinking about convergence of morphological features.
The bat and the bird evolved different bone structures to form a wing. In this case, both (immature) insects and arachnids have the ability to spin silk. Silk has multiple uses, but arachnids commonly use it for trapping and securing prey, whereas insect larvae do not - they use it for spinning a proptective coccoon and sometimes for dispersal. So when we find a caterpillar using silk to trap and secure prey, we are inclined to suspect that this particular use of silk has evolved 'de novo' in that particular lineage becuase it is unique in that lineage. It is a convergence of behavior and silk application made possible by the inherent properties of arthropod silk.
Sorry about the mosquitoes - I just spent a week on the coast up there and had great weather.