There is a great deal of confusion surrounding all of these plants and their classification. I'm pretty sure I've been told in the past, by more than one question setter or asker, that the onion belongs to the lily family. I think this may have been true in the past; but if it ever was, it isn't any longer.
The onion, garlic, leek (etc.), belong to the genus Allium, which is named from the Latin word for garlic. I'm calling it the garlic genus.
The garlic genus, Allium, belongs to the family Amaryllidaceae, which (for equally obvious reasons) I'm calling the amaryllis family. Two other genera in this family are Amaryllis itself, and Narcissus – the latter of which includes (as well as the 'type species') the daffodil. The Amaryllidaceae belong to the order Asparagales, which also (as its name implies) includes the asparagus family (Asparagaceae). The scientific name of the asparagus plant is Asparagus officinalis.
The hyacinth, bluebell and lily of the valley also belong to the asparagus family (Asparagaceae), and the order Asparagales.
The iris family (Iridaceae), which also includes the freesia and the gladiolus, is another member of the order Asparagales; and so is the orchid family, Orchidaceae.
True lilies – which include (for example) the tiger lily, the Easter lily and the Christmas lily – belong to the genus Lilium, in the family Liliaceae and the order Liliales. They are no more closely related to any of the plants mentioned in previous paragraphs than by belonging to the class of Monocotyledons – one of the two groups into which the flowering plants are traditionally divided. (The other is the Dicotyledons – those with two embryonic leaves, as opposed to one). Oh, and by the way: one popular flower that does belong to the lily family is the tulip.
As well as the lily of the valley (and the day lily, another popular garden flower in the asparagus family), the arum lily and the water lily are two more examples of plants that despite their names, are not true lilies; and neither of them is related to any of the other plants mentioned here.
When you hear that lily bulbs are eaten as root vegetables in China, Taiwan and Japan, it only makes all this even more confusing.
© Haydn Thompson 2021