The normal distribution is really a family of distributions. Is the standard normal distribution also...

The normal distribution is family of distributions. When the standard normal distribution is applied consistently across the distribution, it is said to be normally distributed.
The only distribution with a mean of zero and a standard deviation of one is the standard normal distribution.

For instance, z values are often computed using the desired features of the distribution when a particular distribution is believed to be a normal distribution.
So, it is also a family of distributions, according to the conventional normal distribution.

So, yes, the standard normal distribution is also a family of distributions.