Note that the Abaqus manual formula defines the velocity difference, while the formula in the CalculiX manual defines the force. Those are different things.

If you combine the formulae (ignoring the difference in notation) you get:
F₂ = -c·Δv·(x₂-x₁)/L.
That is, the force depends linearly on the velocity difference; which makes sense.

@rsmith I believe we misunderstand each other but can agree on that a common definition of damper force would be F = c·Δv or F = -c·Δv depending on how the local directions are defined so by substitution the Δv from Abaqus will give the same formula as you have written
F₂ = -c·Δv·(x₂-x₁)/L
but in the Calculix manual is written
F₂ = -c·(v₂-v₁)·(x₂-x₁)/L·(x₂-x₁)/L or F₂ = -c·Δv·(x₂-x₁)/L·(x₂-x₁)/L
and I simply cannot understand how you can get this last part "·(x₂-x₁)/L " to disappear from the formula

Where I think you are led astray is that Δv is not simply (v₂-v₁).
As defined in the Abaqus formula, Δv = (v₂-v₁)·(x₂-x₁)/L. Also note that this part is shown within square brackets in the CalculiX formula for F₂ even though that is not strictly necessary.

v and x are vectors, not scalars, so in CCX the force magnitude is F=cΔv and the direction is defined by the unit vector along the line of action (x2-x1/L), the Δv also has to be evaluated in the direction of action, therefore using the dot product of the velocity vector and the unit vector along the direction of action. Same for Abaqus.