Scientifically speaking, a bug is a special type of insect that has mouthparts for piercing and sucking. So technically, insects like flies are not bugs. However, in common language, people would refer to flies and other insects as bugs.

To find common ground between spiders and insects, we have to go back to arthropods, which includes shrimp, crabs, and barnacles, which most wouldn’t call insects or bugs.

I think it’s fine to call most insects bugs, but broadening that to spiders is a bit of a stretch IMO.