Intensive and extensive properties

As you say, flux $bf J$ is the flow of energy generated by a thermodynamic force $bf F$ with some proportionality constant $L$ (conductance), and usually measured as an energy rate per area. This means if you increase the system size at fixed external force, you increase the total amount of energy flow, but the flux stays the same.

From a microscopic view, flux is usually defined as a spatial average of a local current, begin{align} {bf J} (t) = frac{1}{V} int_V {rm d}^3r ~ {bf j} ({bf r}, t)~, end{align} so again you see the normaliztion by system size.

The example with the two suns fits into this picture as they provide the external force $bf F$: when this doubles, also the flux doubles.