Question

But old fears have a way of clinging like cobwebs. What type of figurative language is this?

Answer 1

This is definitely a simile. A way to spot this type of figurative language is to look for like or as. 

Example:

His eyes watched the bird as if he were a predator waiting for the right time to strike.

Answer 2

This is a simile. It uses like or as to convey strong feeling.