I just adopted a unicorn so I will share what I have and haven't done to try and get to the bottom of what you need to get one. I am not friends with a cat or dog, nor have I been. I have a few sim friends, a 8 horse BFF's including the unicorn, and the only small pet I had was a rat who died on me after a few days because I probably forgot to feed it. I did adopt 3 wild horses previous to the unicorn and get the little achievement for that. I also got the charisma achievement for having 10 best friends (pets count towards that).
If I had to make a guess, being friends with "all types" might mean a sim, a pet, and a wild horse (not the unicorn)?