Naw, Mutants are different in that they're just born to be freaks of nature. Ojiro always had his tail, Koda always had his rock-ish face, Tokoyami always had his Dark Shadow. That's why Aizawa's Erasure can't erase the fact that people have too many arms and such; it's just a part of their body. 

Essentially, mutant traits are passed down genetically and such, so things get pretty weird. I'm guessing that any additional abilities will properly manifest at the toddler age, so it's like... tokoyami is born with the bird head, but at the age of 4, he gains Dark Shadow as his 'real' Quirk.