Why does an LED have to be a diode?

The existing answers miss the core of the question.

An LED needs to be a diode, specifically because the way the charge carriers recombine in the forward-biased diode junction releases the correct amount of energy to create photons in the visible range. Passing a current through a chunk of semiconductor with no diode junction in it would simply produce heat.

It's also important for efficiency that the semiconductor be a direct band gap material, so that energy is not lost to phonons (crystal vibrations — heat) rather than photons.

Regular silicon diodes emit light, too, but because the band gap is too low, the photons emitted are in the infrared range, and invisible to the eye. Also, silicon is an indirect band gap material, which greatly reduces its efficiency at producing photons at all.

why does this component need to be a diode to emit light?

By conservation of energy, light emission implies power input. It is normal to deliver electrical power through two wires, so the simplest electric powered light emitter has two wiring terminals, i.e. is a diode.

Two-terminal semiconductors replaced two-terminal tubes (vacuum or gas-filled) having two electrodes, which were called 'diode', and the name has stuck. Electroluminescent panels of yesteryear were also semiconductors that gave off light, but weren't produced in the kinds of high-tech assembly lines that electronic diodes are made in. So, those weren't called diodes.

White "LED" devices around you are not simple semiconductor diodes, but are structures with diodes and phosphors that give off useful amounts of white light, having a blue-emitting diode and red/orange/yellow/green phosphors that convert the blue light. Lenses and other features for effective light emission are common; LEDs do not resemble other practical diodes, except that they have two wires or connecting terminals.

LED stands for Light Emitting Diode; but why does this component need to be a diode to emit light?

Because LEDs are a diode which posses the same characteristics a common solid state diode.

My question assumes that the "leds" we see everywhere (for lighting, screens, etc) are actually diodes -- this assumption might be wrong.

Your assumption is correct.

A diode is an electronic component that has low resistance in one direction. It is a dual electrode (anode and cathode) device where electron flow from cathode to anode is low conductance and the primary electron flow is the high conductance flow from anode to cathode.

The most common diodes are made of crystallized semiconductor materials (e.g. silicone, germanium and gallium arsenide, indium phosphide, sapphire, and quartz) which are doped with p and n type impurities which are separated by the simplest semiconductor building block, the p-n junction.

There are many types of diodes with various characteristics. It is the properties of the p and n dopants and their affect on the voltage-current characteristics of the p-n junction that separates one type of diode from another.

The above applies to all diodes including LEDs.

In LEDs the dopants have electroluminescence properties. When the electrons are crossing the p-n junction, many of the electrons are transformed into sub-atomic particles called photons.

Light Emitting Diodes are called diodes because the are indeed semiconductor diodes that also emit photons in the form of UV, visible light, and IR.