The school system is a wizard's approach to magic. It's analytical, mechanistic, scholarly; no other class should care about it at all. They would have their own ways of organizing spells, which might or might not be reflected in the rules, but certainly should not be subordinated to the wizard system.
A wizard considers fireball and ice storm and concludes they belong in the same group because both are calling up raw energy.
The sorcerer looks at the wizard and says, "Are you nuts? They're total opposites. One is fire and the other is cold. The spells that belong with fireball are Ashardalon's stride and flaming sphere."
The cleric replies, "No, the wizard is right, but the reason is that both are spells of wrath to punish the wicked. Disintegrate is another one in that group. Wall of fire, however, is a spell of guardianship -- I know you both want to put it with fireball, but it doesn't belong there at all."
The druid says, "The sorcerer's on the right track, but needs to look a little deeper. Fireball calls upon the elemental forces of fire; ice storm draws on the forces of water. But Ashardalon's stride mixes the swiftness and suddenness of air with the destructive power of fire. Nothing is purely one or the other. The living world doesn't conform to your neat boxes."
The bard says, "Fireball comes down to us from the Dragon-Mage of Eldamos, she who tamed the seven wyrms. Legend says that she also created cause fear and summon draconic spirit. Now, ice storm is the work of Gorthym, the King of Winter. It's said he was a half-giant, and he made thunderwave when he stomped the earth before battle."
The warlock shrugs. "How many people can I kill with it?"