How to align equations under section name, not numbering?

The "twist" in your setup is that you use roman rather than arabic numerals for the section numbers. This requires setting a separate indentation amount each time \section is run. The following code achieves this by creating a macro called \setmylength, which sets the value of a length variable called \mylength.

\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage{amsmath}

\renewcommand{\thesection}{\Roman{section}}
\setlength\parindent{0pt}

\newlength\mylength
\newcommand\setmylength{\settowidth\mylength{\Large\bfseries\thesection\quad}}


\begin{document}

\section{Euler's Number \textit{e}}
\setmylength

\hspace{\mylength}%
$\begin{aligned}
e &= \lim_{x \to \infty} \Bigl( 1 + \frac{1}{x} \Bigr) ^{x} \\
 &= \sum_{x=0}^{\infty} \frac{1}{x!} \\
e^{x} &= \lim_{n \to \infty} \Bigl( 1 + \frac{x}{n} \Bigr) ^{n} \\
 &= \sum_{x=0}^{\infty} \frac{x^n}{n!} 
\end{aligned}$

\setcounter{section}{27} % just for this example
\section{Euler's Number \textit{e}}
\setmylength

\hspace{\mylength}%
$\begin{aligned}
e &= \lim_{x \to \infty} \Bigl( 1 + \frac{1}{x} \Bigr) ^{x} \\
 &= \sum_{x=0}^{\infty} \frac{1}{x!} \\
e^{x} &= \lim_{n \to \infty} \Bigl( 1 + \frac{x}{n} \Bigr) ^{n} \\
 &= \sum_{x=0}^{\infty} \frac{x^n}{n!} 
\end{aligned}$

\end{document}

---

Addendum to address the OP's follow-up query: If more than one alignment point is needed, replaced the aligned environment with an alginedat environment. E.g.,

$\begin{alignedat}{2}
&e &&= \lim_{x \to \infty} \Bigl( 1 + \frac{1}{x} \Bigr) ^{x} \\
&  &&= \sum_{x=0}^{\infty} \frac{1}{x!} \\
&e^{x} &&= \lim_{n \to \infty} \Bigl( 1 + \frac{x}{n} \Bigr) ^{n} \\
&  &&= \sum_{x=0}^{\infty} \frac{x^n}{n!} 
\end{alignedat}$

Here I have given an answer using nccmath, and scrextend packages with the enviroment

\begin{addmargin}.... \end{addmargin}

I had worked before without your MWE and I think that there are many solutions. With

\begin{addmargin}[.5em]{1em}&<------

you can change the parameters of the left margin of the only equations.

\documentclass[a4paper,12pt]{article}
\usepackage{amsmath,amssymb}
\usepackage{scrextend}
\usepackage{nccmath}
\begin{document}
\section{The Euler's Number $e$.}
\begin{addmargin}[.5em]{1em}
\begin{fleqn}[\parindent]
\begin{align*}
 e &= \lim_{x \to \infty} \left( 1 + \frac{1}{x} \right) ^{x} \\
 &= \sum_{x=0}^{\infty} \frac{1}{x!} \\
e^{x} &= \lim_{n \to \infty} \left( 1 + \frac{x}{n} \right) ^{n} \\
 &= \sum_{x=0}^{\infty} \frac{x^n}{n!} 
 \end{align*}
\end{fleqn} 
\end{addmargin}

\end{document}

I put another example, with \begin{addmargin}[1em]{1em}.

\documentclass[a4paper,12pt]{article}
\usepackage{amsmath,amssymb}
\usepackage{scrextend}
\usepackage{nccmath}
\begin{document}
\section{The Euler's Number $e$.}
\begin{addmargin}[1em]{1em}
\begin{fleqn}[\parindent]
\begin{align*}
 e &= \lim_{x \to \infty} \left( 1 + \frac{1}{x} \right) ^{x} \\
 &= \sum_{x=0}^{\infty} \frac{1}{x!} \\
e^{x} &= \lim_{n \to \infty} \left( 1 + \frac{x}{n} \right) ^{n} \\
 &= \sum_{x=0}^{\infty} \frac{x^n}{n!} 
 \end{align*}
\end{fleqn} 
\end{addmargin}

\end{document}