It seems you might be asking why spreading discrete data has no effect on entropy. Because entropy is a measure of expected surprise, the various labels or values that a thing can take is immaterial. So, the discrete values $x_i$ don't matter, merely their masses and spreading the $x_i$s has no effect.
In the continuous case, spreading things out by scaling inevitably reduces the densities, which affects the entropy as defined in your question. The definition is consistent with our intuition of entropy, Shannon explains, because we typically compare two entropies, and since both are scaled, this effect cancels out. Differential entropy is also consistent with discrete entropy in the sense that it approximates what would happen if the entropy of the quantized distribution were measured.
Note that in the continuous case, spreading things out by other methods can leave entropy unchanged. For example, a uniform distribution over $[-\frac12,\frac12]$ has entropy zero. "Spreading it out" so that it is uniform over $[-10, -9.5], [9.5,10]$ still has entropy zero. Spreading never matters; only the expected surprisal does.
There is one important difference between the continuous and discrete
entropies. In the discrete case the entropy measures in an absolute
way the randomness of the chance variable. In the continuous case the
measurement is relative to the coordinate system [and] the entropy can
be considered a measure of randomness relative to an assumed standard,
namely the coordinate system chosen with each small volume element $dx_1, …, dx_n$
given equal weight. When we change the coordinate system to
$y_1, …, y_n$, the entropy in the new system measures the randomness
when equal volume elements $dx_1, …, dx_n$ in the new system are given
In spite of this dependence on the coordinate system the
entropy concept is as important in the continuous case as the discrete
case. This is due to the fact that the derived concepts of information
rate and channel capacity depend on the difference of two entropies
and this difference does not depend on the coordinate frame, each of
the two terms being changed by the same amount.
The entropy of a
continuous distribution can be negative. The scale of measurements
sets an arbitrary zero corresponding to a uniform distribution over a
unit volume. A distribution which is more confined than this has less
entropy and will be negative. The rates and capacities will, however,
always be nonnegative.
— Shannon 1948