How many professors are children of professors?
I've noticed this too. There is a longstanding literature (mostly in sociology) about social mobility that looks at parents' and children's occupations. Beller and Hout (2006) report that father-to-son occupation correlation was about 0.30 to 0.40, and Piketty (2000) gives an overview of intergenerational mobility for a handbook chapter.

Getting closer to your question, Torche (2011) finds that in the U.S. there is a "U-shaped pattern" of occupational (and other status) similarity, when child education is on the x-axis. For those with just bachelor's degree, their outcomes are barely correlated with their parents', but people with advanced degrees or low education are much more likely to be similar to their parents.

Through Google Scholar, I did not quickly find related literature about whether academic careers stand out from this trend in some way. However, I believe academia would have even higher intergenerational replication, for the following reasons.


First, academic jobs require high levels of formal education, and education is one resource that is often transmitted across generations (cf. Roksa and Potter, 2011). (Professors could especially set up their kids for success while advising on what kind of college to attend, or whether a university offers good undergraduate research opportunities.) 
Further, academia has more or less an apprenticeship model and has many unwritten rules to navigate; growing up with that as the "family business" should help one navigate that career path. 
Finally, while there are lots of doctors and lawyers on TV, and those are professional careers many are familiar with, being a professor is a bit more of a niche career, and it may be harder for people from non-professor families to even think of it early on.