The findings of this study would appear to support an evolutionary explanation of the origin of human speech (as well as music). If humans and birds "have a common biological hardwiring that shapes how they produce and perceive sounds," it could be argued that this "common biological hardwiring" was inherited from some common evolutionary ancestor of birds and humans. However, we might ask why this "common biological hardwiring" appears in birds and humans, who, under standard evolutionary theory, do not share any immediate ancestors, but not in apes, who supposedly do share immediate ancestors with humans. In addition, as Michael Egnor (see here) and others have argued, human language differs in its very nature from the forms of communication used by animals (like birdsong) because it can be used point to things that have no have no physical existence (in other words, abstract ideas). In short, even if it is true that both the ability to produce birdsong and the ability to produce human language are "hardwired," this would not prove that the two have a common evolutionary origin.
Image of a zebra finch from Wikimedia Commons