Ok but “bug” has multiple meanings, and almost nobody means “hemiptera” when they say it. More commonly, it’s any terrestrial arthropod. Arachnids are bugs. Centipedes are definitely bugs.

Heck, there’s a broader definition that basically includes all arthropods. “Moreton bay bugs” are a popular food this time of year. And they’re a kind of lobster.

I make a point of referring to birds as "feather-bugs", much to the weary resignation of my RL friends.
the birds and the bugs
We don’t really know where the phrase came from. My guess is that they’re things from nature that alliterate, which makes it sound cute and innocent.

Birds, mostly males, get all flashy and showy to attract a mate, and bees all answer to the matriarch of the family, so it’s just like life. Obviously.

Maybe the saying came from the mirror universe..