using underbrace without having left( and right) scale

I would suggest using a combination of \vphantom{<stuff>} (for correct vertical spacing) and breaking up the expression into three separate components to correct for the extensible brackets. For example, the following shows the difference between your attempt and using the above combination:

\documentclass{article}
% These definitions are taken from Herbert Voss' Mathmode document,
% available online via CTAN from: http://www.ctan.org/pkg/voss-mathmode
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathllapinternal#1#2{%
\llap{$\mathsurround=0pt#1{#2}$}% $
}
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathclapinternal#1#2{%
\clap{$\mathsurround=0pt#1{#2}$}%
}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathrlapinternal#1#2{%
\rlap{$\mathsurround=0pt#1{#2}$}% $
}
\begin{document}
Here is some text.
\[
  \left(\underbrace{e_{s,i,j}}_{\mathclap{=:x>0}}+\underbrace{e_{t,i,j}}_{\mathclap{=:y>0}}\right)
\]
Here is some more text.
\[
  \left(\vphantom{e_{s,i,j}e_{t,i,j}}\right.% Left bracket
  \underbrace{e_{s,i,j}}_{\mathclap{=:x>0}}+\underbrace{e_{t,i,j}}_{\mathclap{=:y>0}}%
  \left.\vphantom{e_{s,i,j}e_{t,i,j}}\right)% Right bracket
\]
Here is some more text.
\end{document}​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​

This is what it looks like:

You may have to fiddle around with the spacing of the brackets to suit your preference, using math spacing or \mathllap and \mathrlap.

\documentclass{article}
\usepackage{mathtools}

\begin{document}

\[ \big(\underbrace{e_{s,i,j}}_{\mathclap{=:x>0}} +   
        \underbrace{e_{t,i,j}}_{\mathclap{=:y>0}}\big) \]

\end{document}​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​

I prefer \smash[b]:

\documentclass{article}
\usepackage{amsmath}
\usepackage{mathtools}

\begin{document}
Here could be text.
\[
\left(2,\smash[b]{\underbrace{\frac{5}{3}}_{\mathclap{\approx 1.66666667}}} \right)
\vphantom{\underbrace{\frac{5}{3}}_{\approx 1.66666667}}
\]
Here could be more text.

\end{document}

because it would also work together with \DeclarePairedDelimiter (see https://tex.stackexchange.com/a/435707/128042):

\documentclass{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{mathtools}
\usepackage{hyperref}
\DeclarePairedDelimiter{\norm}{\|}{\|_{\href{https://de.wikipedia.org/wiki/Euklidische_Norm\#\%E2\%84\%932-Norm}{\ell_2(\mathbb{R}^2)}}}
\begin{document}
Here could be text. 
\[
\norm*{\left(2,\smash[b]{\underbrace{\frac{5}{3}}_{\mathclap{\approx 1.66666667}}} \right)}
\vphantom{\underbrace{\frac{5}{3}}_{\approx 1.66666667}}
\]
Here could be more text.

\end{document}

\smash[b] is especially useful, if you have more complicated delimiters defined in the preamble like colored or linked delimiters (see https://tex.stackexchange.com/a/360871/128042)

A short description of \DeclarePairedDelimiter can be found here: Easily change behavior of \DeclarePairedDelimiter

A detailed description of \smash[b] here: https://tex.stackexchange.com/a/424204/128042