For v4... (I think it works for v4.5 since it is almost the same thing)
You have to wait for it to be 3 years old (aka, 3 earth days, 72 hours) Then, when it turns into something, you MAY get a special charcter. I think your problem is, your in a certain family, (example, you want the mametchi special charchter, you need the "350" in there) but you have the most skill in one certain skill, but in the wrong family. (example: You want the mametchi family special charcter, so you get all that you need in that skill. But your not in THAT family (your not in the mametchi family, you're in some other) so you can't obain the special charcter.)
You also need to be in the same gender as it, and that's a possibilty.
(yes, it happened to me, except I just wanted a lot of skill)
If you don't get the "family" thing go here:
https://www.tamatalk.com/IB/index.php?showtopic=99281
I hoped that helped you,
mametchi
p.s. You'll need to be at least 6 to have the matchmaker visit you!