How is this not a counter-example to the fact that the standard deviation of a standardized random variable must be 1?

Question

Let $X$ be a random variable that sends all members of the sample space to some constant real number $c$. Let $Z$ be its standardized counter-part.

The standard deviation of $X$ is $0$, since $X$ is constant. The standard deviation of $Z$ is also -- it seems -- $0$.

But doesn't this contradict the well-known fact that the standard deviation of a standardized random variable must be $1$?