Only Child Syndrome?

So is this "Only Child Syndrome" a real thing? I have a sibling, and older sister. I grew up with the typical sibling rivalry. Wanting to do everything she did, go where she went, annoying little sister. My sister is probably reading this, nodding in agreement. I was the loud one, she was the quiet one, can you guess who always got in trouble? But I was never lonely, I always had someone to play with even when she didn't want to! I have no idea what it's like to be "only."

My daughter has been asking for a sibling since she was 4. The notion died out for a bit when we explained she would have to share everything, ha. But creeps back, periodically. We tried for a while, even went the IVF route(story for another day), was unsuccessful. So here we are, only child.

I feel like my whole world is this child. Check my iPhone photo album. It use to be my puppies and selfies, now I have 5k photos of her. So she is my everything, I know this, and it's probably apparent to everyone else, but....., well to her, I'm just mom. I'm the rule maker, the protector from bad dreams, the hair dresser, make up artist, cook, maid, tutor, vocal coach, etcetera, etcetera. I try to play games with her, but with all those jobs, hard to find time. I don't have time to play with dolls when I'm running my own business, working a full time job, singing in a band, and managing her acting. Whew...I'm pooped when I get home each night, then get her ready for bed, make sure she had dinner and do her Spanish lesson, read and Khan Academy. I'm not complaining, my favorite job is being her mom. She tells me she is bored, or doesn't want to play in her room by herself. We live quite a distance from any school friends so playdates have to be arranged.

Am I failing at socializing her more? She is super outgoing, makes friends easily, but as I have made her my world, I feel like I inadvertently have her believing she is everyone's whole world. She doesn't share, she hates being second, and has to be the center of attention. Gah! Have I given too much attention? Too many gifts and activities to fill the void of a sibling? What can I do reign her back in? They say, "some gotta feel before they fix." Can I help her recognize and appreciate what she has and that she doesn't get to have everything her way? Send thoughts and ideas my way!