I had a ton of bffs of all kinds (sims, horses, wild horses, dogs, cats, strays) and I still couldn't get my bff unicorn to move in. After I read some of the posts here I thought, "Maybe I should increase the number of pets I have," and I adopted one more, bringing my pet count to three. Success!
So in an effort to further understanding:
1 familiar horse
2 human bfs
1 wild horse bf
1 horse bf
1 stray cat bf
1 dog bf
2 horse bffs
3 pet horse bffs
And none of it counted untill I adopted the third horse. (It was the fourth wild horse I adopted, so it might be the wild life skill. Is there any way to tell what level it is?)
Oh, and my riding skill was maxed, and I still got the message.