[This is an awesome question! I have never thought about this before and truly enjoyed doing my research on this! Just so you know, THIS is the kind of question that gets a teacher of Literature very excited! This being said, my answer follows.]
Atticus truly is given finch-like characteristics, all of them positive. Although most resources tend to stick with logistics about genus and species in regards to finches, I found a wonderful website (a subdivision of Netpets) that focused on "finch personalities."
First, the most common type of finch (often called "Type 1") is known to have a "gentle personality" and be "good community types." For goodness sake! Yes, Atticus has a gentle personality: shown in no better instance than how he has always parented his children. As for being a "good community type," can there be a better philanthropist than one who defends the poor and downtrodden in a court of law to achieve justice?
Finches "get along well with other gentle types, but may be hurt by tougher birds" (often called "Type 2" and "Type 3"). Doesn't your mind go immediately to Ewell trying to hurt Scout and Jem while Boo Radley saves them?!? Now listen to this:
For a large aviary with many hiding places, it might work to place [the Type 1 finch] with the tougher birds.
I was floored yet again to read this description of finches. It made me think of the sanctuary that is Atticus' home where he raises his children: Atticus' "hiding place." Atticus is truly placed with "tougher birds" and makes his own "aviary" (the town of Maycomb) a better place as a result.
A final aspect of finch personality reported by Netpets is to try and avoid crowding in order to offer privacy. My mind immediately went to Atticus in front of the jail, keeping peace for Robinson from the angry mob.
Oh, dear Atticus Finch, sing your song of equality for us and erase the tougher birds from Maycomb and our universe!