I had a cake made like this for me when I was 5 so wanted to do the same thing for my daughter. It was for her 4th birthday.
I did not use a special pan. I used a boxed cake mix and cooked it in a large glass mixing bowl. After came out of the oven I took it out of the bowl and cut a hole in the middle for Barbie and put her in the hole.
I then piped frosting on to the cake. I made a butter cream frosting and the regular star tip that comes with the piping bag.