No, and not even close. Consider a tree of life. Formally, vertebrates are subphylum Vertebrata within the phylum Chordata (the chordates). That already eliminates most of the animals, as most animals are invertebrates – defined as being outside Vertebrata. All the arthropods, molluscs, echinoderms (like starfish), sponges, cnidarians {jellyfish, corals, anemones, etc.), nematodes, and so on are invertebrates.

Instead of just assaulting you with terminology, here’s a phylogenetic tree showing the vertebrates within the chordates and the chordates next to our nearest neighbors.

To get fish from the vertebrates, you simply exclude the tetrapods (clade Tetrapoda), which are the four-legged vertebrates. However, even within the tetrapods, plenty of tetrapods also have no legs (snakes, for example); they’ve simply lost them through evolution, a phenomenon called “secondary loss”.

Here’s a phylogenetic tree, meanwhile, within the vertebrates. Anything outside “Tetrapoda” on that tree is a fish.

TL;DR: There are many, many, many legless animals that aren’t fish.