This is an important question because it highlights the limitations of a metaphor. Indeed, Boo Radley in To Kill a Mockingbird is rightly associated with the mockingbird. Scout makes the connection when she realizes that to subject Boo Radley to the attention of the community and the praise for his rescue of Scout and Jem would be something like killing a mockingbird. Boo has been trying to spread beauty in the world, just like a mockingbird does with its song. He has not harmed anyone (other than Ewell, of course, but that is to save the children). Scout realizes that Boo's spirit would be crushed, even killed, by attention he does not want and cannot handle.
The metaphor, however, goes only so far, for Boo Radley is not completely like a mockingbird. For one thing, he does not spread his song of beauty throughout the world for all to hear. He is, in fact, a recluse, and he appears to be a highly anxious person who is extremely uncomfortable around other people. Even with only Atticus, the children, and Mr. Tate, Boo holds back, unable to cope with the situation until Scout steps up to guide him. Then he follows her lead.
Boo's inner beauty and love, then, are only known to a few people. He chooses to share them with Scout and Jem, first by leaving little presents for them in the hollow tree. Then he stitches up Jem's pants for him. He clearly watches out for the children as well, even though they don't know it, and when they are in danger, he rushes to the rescue and saves them from what might have been severe injury or even death at the hands of the vengeful Ewell. Boo's sphere of influence, unlike that of a mockingbird, is limited, yet it is still powerful and has a profound impact on the lives of Scout, Jem, and Atticus.