Simple case
It is well-known that if we have a regular hexagon on a plane, then every line that passes through the centre of the circumscribed circle bisects the area of the hexagon.
Extension
Assume we have a (2D) regular hexagon - like fractal, like the one in the image.
My question is: Can we claim the same as above? In general, I guess there is some dependence on symmetries, but I am not quite familiar with the whole concept.
Yes. Notice that your figure, as well as hexagons themselves, are centrally symmetric - that is, reflecting through the center point yields the same image. Thus, if you choose any line through the center splitting the plane into an upper and a lower section, the reflection through the center takes the upper section to the lower section and vice versa - thus the area of the hexagon in each must be equal.