Did baby bird hair and mammalian hair convergently evolve?
Bird "baby hair" are modified feathers.  (Or perhaps feathers are modified "baby hair".  The ancestral state for feathers was probably closer to the chick's down in appearance, but I don't know whether chick down is unmodified original feather or whether it was feathers and then modified again.)

In any case, mammalian hair and dinosaur feathers (which includes bird feathers) are separately evolved from the same base structure, modified reptilian scales. They are therefore homologous, but evolved independently.  


  These results reveal a new evolutionary scenario where hairs, feathers, and scales of extant species are homologous structures inherited, with modification, from their shared reptilian ancestor’s skin appendages already characterized by an anatomical placode and associated signaling molecules.


--The anatomical placode in reptile scale morphogenesis indicates shared ancestry among skin appendages in amniotes