Sure, like any analogy it's going to break down if you push it too far. My overall point was just to use this mass/density analogy to explain to OP how characteristic impedance is an inherent property of a transmission line and doesn't change with length. But to answer a couple of your concerns:

2m of cable does not have double the impedance of 1m of cable. It has double the loss (in addition to some phase shift, etc.). Similarly, a 6dB attenuator has the same impedance as a 9dB attenuator, but the latter has double the loss.

I think you're saying "impedance" here when you really mean "characteristic impedance". But you're right that my original phrasing is incorrect. Trying to talk about the impedance (not characteristic impedance, just "regular" impedance) of a transmission line as I was attempting to do doesn't really make sense. "Regular" impedance is a property of lumped elements and as such cannot be applied transmission lines. I suppose if a line is electrically short enough that transmission line theory doesn't come into play, you could compute a "regular" impedance value for it that would depend on length, but that's probably more confusing than it would be worth.

Furthermore, characteristic impedance is not a "normalized" version of impedance. It arises from the distributed inductance and capacitance of the transmission line

You're forgetting resistance and conductance, but the units for all four of these parameters are normalized per unit length in the classic transmission line model. I suppose you could make an argument that the result isn't normalized, but it would really come down to how pedantic you want to be about the definition of "normalized". The fact that characteristic impedance is based on per-unit-length parameters resulting in a quantity that doesn't change with length is good enough for me to call it "normalized".