Align math to the left

That's a rather strange requirement, perhaps easiest is if you use $$ instead of \[. $$ is officially not supported syntax, the main problem being that it does not+ work with fleqn which seems to be exactly what you want here.

Note never leave a blank line above a display math, whether $$, \[ or \begin{align}

\documentclass[fleqn]{article}

\usepackage{amsmath}

\begin{document}

Classic math which I don't want align to the left
$$
    2x + 5
$$

Multiline math which must be left-aligned   
\begin{gather*}
    x^2 + 3 \\
    2x - 5 \\
\end{gather*}

\end{document}

You can do that nesting gathered in aflalign(*) environment:

\documentclass{article}

\usepackage{mathtools}
\usepackage[showframe]{geometry}

\begin{document}

Classic math which I don't want align to the left

\begin{equation}
\begin{gathered}
    x^2 + 3 \\
    2x - 5 = z
    \end{gathered}
\end{equation}

Multiline math which must be left-aligned
\begin{flalign}
\hskip\parindent & \begin{gathered}
    x^2 + 3 \\
    2x - 5 = z
    \end{gathered} &
\end{flalign}

\end{document}