This isn’t normal, and its strangeness is meant to be a part of the characterization in the novel.

A few points about the characters:

-Scout (and Jem) have a close and affectionate relationship with their father.

-Atticus is pretty unconventional in how he approaches things. He probably “looks the part” of an old-style aristocratic southern politician (he’s a state assemblyman as well as an attorney). But in his day-to-day personal life, he’s not all that strict about maintaining tradition. 

-he became a father somewhat older, and his wife is deceased. So the children probably hear other people call him “Atticus” or “Mr. Finch,” and their mom isn’t around to refer to him as “dad” to the kids. 

-the dynamics of social class are almost as important as those of race in this novel. The relative security of a character tends to correlate to better behavior. (I’m not justifying this, but it does seem to be the case in the book: the violent racist lynch mob-forming Cunninghams and the child molesting racist Bob Ewell are all lower class. The upper class characters are depicted much less negatively.) Atticus is secure in his position, and tolerating his children calling him by his first name is probably part of his generally calm, aristocratic attitude. Forcing his children to call him “dad” (or whatever) and punishing them for using his first name would probably seem like trashy behavior to him (compare the fact that his concern with his children’s use of the n-word is that it’s “common,” or vulgar, not that it’s hurtful.)

This is all kind of “reading between the lines,” because Scout’s use of “Atticus” isn’t fully explained in the text. But I do think it’s deeper than just using his first name to the reader. She does seem to call him this to his face.