Phantom equals sign in aligned equation

Your first attempt almost had it: you just need to write \phantom{{}={}} instead of \phantom{=}.

In the picture below, I've inserted Xs at the start of each line to help ascertain visually that the alignment is working.

\documentclass[preview]{standalone}
\usepackage{amsmath}
\begin{document}
    \begin{align*}
        &\phantom{{}={}} x^2 - 1 \\
        &= (x - 1) (x + 1) \\
        &= 7
    \end{align*}
\end{document}

Rather ugly, but if you like...

\documentclass{article}
\usepackage{amsmath}

\begin{document}
    \begin{align*}
        {}& x^2 - 1 \\
        ={} &(x - 1) (x + 1) \\
        ={} &7
    \end{align*}
\end{document}