This is not true in general. E.g., let
$$A:=\left(
\begin{array}{cc}
-i & 0 \\
0 & 0 \\
\end{array}
\right),\quad A':=\frac i2\,
\left(
\begin{array}{cc}
-1 & 1 \\
1 & -1 \\
\end{array}
\right).$$
Then, for real $t$,
$$e^{-tA}=\left(
\begin{array}{cc}
e^{i t} & 0 \\
0 & 1 \\
\end{array}
\right),\quad
e^{-tA'}=\frac12\,\left(
\begin{array}{cc}
1+e^{i t} & 1-e^{i t} \\
1-e^{i t} & 1+e^{i t} \\
\end{array}
\right),$$
and
$$\|e^{-tA}-e^{-tA'}\|=\sqrt{1-\cos t}, $$
so that
$$\sup_{t\in\mathbb R}\|e^{-tA}-e^{-tA'}\|=\sqrt2\ne2(1-\delta_{A,A'}).$$