She was put with Cadence and Shining Armor, as it was believed good parents would save her. Turns out that that would have its own struggles that Twilight couldn't predict. Cozy Glow has a condition where she simply doesn't age, a condition possibly Bat Pony related, but who knows really? She was an orphan discovered by a Bridleway manager and raised to be little more than a child icon.
Upon adoption, Cozy realized that she had truly missed out on family and parents, and through her little "sister" Flurry Heart, she would see her own motherhood that she would never experience due to her condition. It was therapeutic in a way, until Flurry Heart grew to twice Cozy's size. That bish. But maybe…things with the Princess of Love and Prince of the Crystal Empire wouldn't be so absolutely horrible.