The power to bend the five elements (air, earth, fire, water, and energy) is linked specifically to the nations (and people) of the Avatar World by an as yet unknown mechanism. With the exception of the Air Nomads, mixing of bending types does not necessarily ensure what bending abilities will manifest, or whether ANY bending ability will be inherited at all. Nor is it clear how the power moves between family members. Even twins of parents from two different lands can possess different bending abilities.
Since there are variant powers such as metal-bending (a variant of earth-bending, or explosion bending (variant of fire-bending), this might also denote why there is such a study of lineage among the people of their world, to attempt to predict what bending abilities and strengths might manifest in a family line.
From the Avatar Wikia article on Bending Arts:
Though each nation is affiliated to a specific bending art, not all
from any nation, with the exception of the Air Nomads, are born with
the ability to bend. It is unclear how the ability to bend is
distributed amongst the population or how closely it is linked to
heredity; even among identical twins it is possible for one to be a
bender and the other not. In the case of mixed marriages, the children
may be benders of either element if they are benders at all; for
example, firebender Mako and earthbender Bolin are brothers with
parents from the Fire Nation and the Earth Kingdom.