Also, note that coupling the verb with "moan" softens it. "To bitch about" by itself is considerably stronger than "to bitch and complain" or "to bitch and moan."
Yes, the "bitch and moan" is quite a bit softer. The very unpleasant association of the word "bitch" to descibe any woman a person doesn't like makes using only "bitch" (even as a verb) quite a bit nastier in sound. I wouldn't use THAT, but "bitchin' and moanin' about..." just means to complain.
Children are very good at knowing what they can do and what they can't, with which audiences. It's even possible when they are at their father's house, a different vocabulary is used than when they are at mine. I get much more upset over one of them calling the other "stupid" or "an idiot" than I would about a word generally considered a bit vulgar.
What's accepted as vulgar does vary by the audience, but mean names are the same everywhere.