milk, woman's breasts, nurse
奶 centers on the nurturing, life-giving role of a woman: milk, breasts, and the act of nursing or caring for an infant.
奶 combines 女 (woman) with 乃, which likely acts as a phonetic component. The character is associated with the female breast and nursing.
A woman 女 with a breast-like shape 乃 beside her: 奶 represents the milk and nursing care a woman provides.
For ダイ, picture a mother nursing her baby while saying 'die' (as in 'that's my little die-die' — a playful invented sound for a baby). The sound 'die' cues ダイ, and the scene ties to the nurturing meaning.