The story
Blend of two + teenager (by analogy with threenager), due to toddlers of this age ostensibly resembling the difficult attitude and demeanour stereotypically associated with teenagers.
Etymology from Wiktionary, CC BY-SA 4.0.
twonager
Today it means: A two-year-old toddler who is now more independent and difficult, involving refusals, demands and tantrums; a toddler experiencing their terrible twos developmental stage. Full definition →