I think all the answers given so far are useful, but I'd like to comment on your question:
We sometimes see children who look very much like their father or mother, or even behave typically like either of them.
I'm not sure how well English expressions cover both of those circumstances. There are hereditary traits (which would include physical characteristics, such as hair and eye color, skin tone, and even whether or not you might be genetically predisposed to certain diseases, such as diabetes) and then there are behavioral traits (which might involve work ethic, values, and religion).
In other words, there are two ways to pass on something from generation to generation: through genetics, or through learned or emulated behavior.
As I mentioned in my comment to Andrew, there's the idiom the apple doesn't fall far from the tree, but I usually hear that in a negative context when it relates to behavioral traits, or else neutral contexts when it's applied to physical characteristics:
"I heard Imelda went into rehab again."
"Not surprising; apple doesn't fall too far from the tree."
On the other hand, chip off the old block can be used more positively:
"After getting his meteorology degree last June, Greg landed a great job with NOAA."
"Wow, he's a chip off the old block!"
If I overheard those two conversations, I'd assume that at least one of Imelda's parents had dealt with substance abuse problems, and that Greg's dad had some kind of technical degree, and was involved in atmospheric research or something similar. (Incidentally, the fact that I mentioned Greg's father, as opposed to his mother, doesn't reveal a sexist assumption, as if I'd be surprised by a woman scientist. Rather, it's an assumption based on the idiom used; that particular idiom is usually reserved for father-son characteristics. I'm not entirely sure about this, but I believe that particular idiom and the relatively common nickname of "Chip" for a young boy are related.)
However, if we were discussing skin and eye color, or a certain shape of nose, I'd be more likely to simply call these hereditary traits, or say something like "that's passed down genetically." Of course, this all depends on context, too. I'd be much more careful about how I used words like genetic and hereditary in a journal than I would be in casual conversation. Sometimes words like these are less interchangeable when used within the scientific community, as opposed to the general population.