I actually had a co-worker with a two-year-old, nameless daughter. Every time my colleague suggested something to his wife, she’d accuse him of cheating on her with a woman by that name (yes, the wife was mentally unstable). The couple referred to the girl as “sweetie” or “baby.”
Co-worker brought the little girl to an office softball game. Someone, unfamiliar with the sitation, asked her, “What’s your name?” The group of us froze in discomfort, expecting drama to ensue. Instead, the girl smiled and said “Allie.” I have no idea how the child got named; I’m guessing that her father had had enough and handled things unilaterally.