For a long time, The Kid adopted to this pretty well. I've been Papa in this house for the 10 months she's been actively talking, until lately. She hears Daddy a lot at daycare, where I am the only parent who prefers Papa. All of her classmates refer to their father's as Daddy, so I shouldn't be shocked she's picked it up.
The transition has been gradual. For a time, when she would be excited and she couldn't decide who to tell first, The Kid would refer to us as "MaPa." It was cute when she would say it, but she has since figured out how to get the correct name out when she's excited. Then we went to "DaddyPapa," which is a mouthful.
I realized Tuesday that she has fully transitioned to "Daddy," when I picked her up from daycare. Toward the end of the day, as the toddlers go home, they have the last few kids playing together in one room. The teacher in that room heard me ring the doorbell, so she looked out and I saw The Kid's head peer out the cracked door. As I walked in, she shrieked, "DADDY" and bolted through the door to run to me.
Wednesday was a similar scenario. She was the last toddler in the building, so she was sitting at the front desk with her teacher. As soon as I walked through the door and The Kid locked eyes on me, she jumped down from the chair she was on and yelled, "DADDY!"
She's also added a "you're so silly" version of Daddy, as well as an angry tone for Daddy.
The only reason I'm cool with being Daddy is The Kid is cool with it. She knows who I am, and is still excited to see me everyday. Which brings me to my point, titles are one thing, but it's the role I'm playing that counts.