Consider a young, single woman who immigrates to the USA and then becomes a single mother with two children and qualifies for public assistance. For #1 the household is counted once. The contribution to the Y value is 1 to the numerator and 1 to the denominator. For #2 the family counts for three persons – the mother as one immigrant and the children as two natives. The contribution to the value of Y for natives is 2 to the numerator and 2 to the denominator. Obviously such counting increases the overall value of Y for natives, enough overall to make little difference between immigrants and natives.
Clearly a US citizen who favors open immigration would tend to choose #2 as evidence to support his/her opinion, and one who favors strict immigration controls would tend to choose #1 as evidence to support his/her opinion.
Household statistics seem more objective. Adults earn most income, obtain public assistance, and pay the bills, not children.