Yep. The unicorn spawns in every worl. They also spawn in custom worlds as long as you have fishing venues. In my world, the unicorn spawns about 75% of nights. I don't know if this is different then the pre made worlds, but it's not too bad.
My world I created:
Has a ton of Unicorn spawners Willow Harbour was a special request equestrian world I built for a friend. She loves horse so much, I figured to add a bunch of "Horse Crossing" ,and "Unicorn Crossing" signs/spawners. Every night a unicorn appears for some, and for some others, they have more than three wild horses....lol
It is not a guarantee these instances will happen to you, however they are more likely with the world of Willow Harbour
Just figured I would put my two cents in to see if they are worth anything to anyone