Sodium has greater density than lithium, while most other materials used in a battery have similar densities regardless if sodium or lithium is used, so if a Na-ion battery and a LFP battery have about the same mass and stored energy, it is likely that the sodium-ion battery has a smaller volume.
that doesn't check out, capacity depends on surface area, if the element that is on the surface is heavier then, all other things equal, the battery will be heavier for same kWh.
Sodium would need to be more efficient to be lighter, which it isn't
The maximum deliverable power depends on electrode area, through the maximum current density.
The capacity of storing energy does not depend at all on area, but only on the mass of sodium contained in the battery and on the efficiency of using it (i.e. between full discharge and full charge not 100% of the sodium or lithium is cycled between the 2 oxidation states, but a fraction, e.g. 90%).
Any battery has both an energy density and a power density, which are weakly correlated and the correlation may have opposite signs, i.e. for some batteries it may be possible to increase the power density if the energy density is lowered and vice-versa.
For a given stored energy in kWh, the required mass of sodium is several times greater than the corresponding mass of lithium, by a factor that is the product of the atomic mass ratio with the ratio between the battery voltages. The voltages are similar, with a slight advantage for sodium, so the required mass of sodium is about 3 times the corresponding mass of lithium.
If the complete batteries have about the same mass, that means that other components of the sodium-ion battery are smaller and/or lighter.
It is all about cost and efficiency... There is a classic 1913 electric vehicle that ran NiFe packs for many years, and were only replaced because the container rotted away. Sustainable storage costs real money, but has existed for over a century. =3