To construct a level $\alpha$ rejection region we first calculate the level $\alpha$ critical value $c_\alpha$. For a two-tailed
test based on a test statistic that is $N(0,1)$ under $H_0$, the critical value is defined implicitly by
\begin{equation}\label{deficritval}\tag{1}
1-\alpha/2=\Phi(c_\alpha)
\end{equation}
where $\Phi$ denotes the standard normal CDF. Hence,
$$
\Phi^{-1}(1-\alpha/2)=c_\alpha
$$
where $\Phi^{-1}$ denotes the quantile function.
The probability that $z > c_\alpha$ is $1 - (1 -\alpha/2) = \alpha/2$, and likewise, $P(z < -c_\alpha)=\alpha/2$, by symmetry. Thus, $P(|z| > c_\alpha)=\alpha$, as desired. For example, when $\alpha = 0.05$, $\Phi^{-1}(1-\alpha/2)= 1.96$.
The $p$-value is defined as the smallest level for which a test based on an obersved statistic $\hat{z}$ rejects.
For a two-tailed test,
$$
p(\hat{z}) = 2(1- \Phi(|\hat{z}|))
$$
To see this, note that the test based on $\hat{z}$ rejects if
$$|\hat{z}| > c_\alpha$$
This is equivalent to
$$\Phi(|\hat{z}|) > \Phi(c_\alpha),$$
because $\Phi$ is strictly increasing.
Further, from eq. \eqref{deficritval}
$$
\Phi(c_\alpha)=1-\alpha/2
$$
The smallest value of $\alpha$ for which the inequality holds is thus obtained by solving the equation
$$\Phi(|\hat{z}|) = 1-\alpha/2$$
for $\alpha$, which gives $2(1- \Phi(|\hat{z}|))$.
Hence, we require that the test statistic be $N(0,1)$ under the null and that we reject for both very negative and very positive values of the test statistic (i.e., conduct a two-tailed test).
Whether the result applies to Student's t-test therefore depends on the null distribution you entertain. If you can make a normality assumption on the data (see e.g. here to what that refers more precisely in a regression context) to which you apply the test, it is well known that the t-statistic follows a t-distribution. Hence, you would need to replace $\Phi$ with the corresponding c.d.f. of the t-distribution.
On the other hand, even without a normality assumption, the t-statistic will usually be normally distributed in large samples thanks to a central limit theorem. See e.g. here.
