What any mathematical object means "physically" depends entirely on how you are using mathematical objects to represent "physical" things.
For example, if $c \delta(x)$ it shows up as a term in energy density, then it contributes $c$ to the amount of energy contained in any region containing the point $x=0$.
Also what is the point of scaling $\delta(0) = \infty$ by a constant?
The equation $\delta(0) = \infty$ is* a fabrication -- it's a white lie meant to avoid concept of generalizing the notion of function.
(many -- myself included -- would argue the lie does more harm than good, but that's another topic)
The Dirac delta only has meaning inside an integral -- or in other situations related to or derived from this meaning, such as appearing in a (generalized) differential form -- and that meaning is $ \int_{-\infty}^{+\infty} f(x) \delta(x) \, dx = f(0) $ or some variation thereof, depending on technical concerns and convention. How $c \delta(x)$ differs from $\delta(x)$ is now obvious, by seeing how it affects the integral.
*: For the sake of honesty, I should point out that for some generalized notions of function, there is a corresponding generalized notion of evaluation in which the equation is true. But if you can understand the meaning of that, you wouldn't be confused by $c \delta(x)$.