Aren’t spiders insects?

No. Insects have 6 legs, spiders have 8.

Are they in some separate category?

They’re separate biological classes.
So they’re about as far apart as you are from a reptile, bird or fish.

Not exactly. Humans, birds, and reptiles are all within the phylum chordata, while arachnids and insects are both within the phylum arthropoda.

Fish, interestingly, aren’t a real thing in terms of formal classification. The term is similar to bug in that we apply it to whichever creatures we feel fit the description.

Fish, interestingly, aren’t a real thing in terms of formal classification

This is misleading. Formal classification existed for a long time before phylogenetic classification became the standard.

Pluto used to be considered a planet, but I’m not going to tell people it is one today. Pisces as a class was abandoned due to the realization that we were mistaken about how similar/related they are to each other. Whales used to be included in pisces.