The colon does not align with the \vdots

I don't think these should align on the dots of :=, but rather should be centered with respect to either the s_k or perhaps the relation :=. The dots in := serve a different function to those in \vdots. The package mathtools provides the command \vdotswithin{...} for centering such \vdots with respect to the material ...; it also provides \coloneqq for a better version of :=:

\documentclass{article}

\usepackage{mathtools}

\begin{document}

\begin{align*}
  s_{1} &\coloneqq\sum_{1 \leq j \leq m} X_{j} \\
  s_{2} &\coloneqq\sum_{1 \leq j<k \leq m} X_{j} \cdot X_{k} \\
  \vdotswithin{s_{k}}& \\
  s_{k} &\coloneqq\sum_{1 \leq j_{1}<j_{2}<\cdots<j_{k} \leq m}
          X_{j_{1}} \cdot X_{j_{2}} \cdot \cdots \cdot X_{j_{k}} \\
  \vdotswithin{s_{k}}& \\
  s_{m} &\coloneqq X_{1} X_{2} \cdots X_{m}
\end{align*}

\begin{align*}
  s_{1} &\coloneqq\sum_{1 \leq j \leq m} X_{j} \\
  s_{2} &\coloneqq\sum_{1 \leq j<k \leq m} X_{j} \cdot X_{k} \\
        &\vdotswithin{\coloneqq} \\
  s_{k} &\coloneqq\sum_{1 \leq j_{1}<j_{2}<\cdots<j_{k} \leq m}
          X_{j_{1}} \cdot X_{j_{2}} \cdot \cdots \cdot X_{j_{k}} \\
        &\vdotswithin{\coloneqq} \\
  s_{m} &\coloneqq X_{1} X_{2} \cdots X_{m}
\end{align*}

\end{document}

Welcome to TeX-SE! It does not align because a colon is a relation, i.e. of the \mathrel type. If you want to align it "by hand" you could just make the \vdots of that type, too.

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\[
\begin{aligned}
s_{1} & :=\sum_{1 \leq j \leq m} X_{j} \\
s_{2} & :=\sum_{1 \leq j<k \leq m} X_{j} \cdot X_{k} \\
      & \mathrel{\vdots} \\
s_{k} & :=\sum_{1 \leq j_{1}<j_{2}<\cdots<j_{k} \leq m} X_{j_{1}} \cdot X_{j_{2}} \cdot \cdots \cdot X_{j_{k}} \\
      & \mathrel{\vdots}\\
s_{m} & :=X_{1} X_{2} \cdots X_{m}
\end{aligned}\]
\end{document}

My interpretation of changing of the answer given by the user @marmot using:

• The macro \coegual instead of := simply because it provides the perfect alignment between : and =;
• Using the negative space with the command \mkern I tried to bring the products of the elements close to the summation because I do not like them distant;
• I have used \substack command because the subscripts on the summation did not bring me the products far away.

---

\documentclass[a4paper,12pt]{article}
\usepackage{amsmath,amssymb}
\newcommand{\coegual}{\mathrel{\mathop:}=}
\begin{document}
\[
\begin{aligned}
s_{1} & \coegual \sum_{1 \leq j \leq m} X_{j} \\
s_{2} & \coegual \sum_{1 \leq j < k \leq m} \mkern-5mu X_{j} \cdot X_{k}  \\
      & \mathrel{\vdots} \\
s_{k} & \coegual 
\sum_{\substack{1 \leq j_{1}< j_{2} <\cdots\\ \cdots< j_{k} \leq m}} \mkern-15mu X_{j_{1}} \cdot X_{j_{2}} \cdot \cdots \cdot X_{j_{k}} \\
      & \mathrel{\vdots}\\
s_{m} & \coegual X_{1} X_{2} \cdots X_{m}
\end{aligned}
\]
\end{document}