I am not the child's father, but was with his mother through out her pregency and until a week ago when she left. The baby is 2yrs and 5 months. He believes I am his dad and calls me DaDa. Almost everybody believes he is my baby and we live together for over 3 years. Do I have any legal rights regarding this child

In a nutshell - no. I'm sorry but that is how it works.

Actually... it depends on where you are and what you've done already.

Did you sign the birth certificate or affidavit of paternity? Were you married when the child was born?