I think you've answered your own question. Shyness, like extroversion, has a genetic component. That doesn't mean that you can't expect to influence your daughter's behavior. What you and others do will always have some influence on her development. But it is usually a mistake to think that we can completely reverse behavior tendencies such as shyness. Certainly she doesn't sound like she has any learning problems, but she is obviously shy. And she may remain that way all her life.
In your question you didn't say whether you got over your shyness. If not, you know to expect her to continue the way she is. If so, you might reassure yourself that she will grow out of it somewhat, if not totally. I wouldn't try to do anything special to get her to change this behavior. Reassure the family that what she is demonstrating is shyness, not slowness. And tell her daycare teacher that your daughter talks and sings freely at home, demonstrating things she has learned during the day.
Your daughter is too young for any speech therapy, which she doesn't appear to need if she talks to you at home, or psychotherapy, should this non-communicativeness be due to emotional problems. Just continue to love her, talk and read to her and encourage her to talk to you. Teach her some little songs and then sing them with her; don't expect her to want to sing them as solos.
Our parenting advice is given as suggestions only. We recommend you also consult your healthcare provider, and urge you to contact them immediately if your question is urgent or about a medical condition.