How to place single comma after two equations so it's placed between two strings

You can use aligned to generate an align-like block, and then place the conditions after it. I don't particularly like the placement of ,, so suggest that you remove it since it makes sense without it. Alternatively one can use an rcases environment (from mathtools), perhaps.

\documentclass{article}

\usepackage{mathtools}

\begin{document}

\[
  \begin{aligned}
    f(x) &= \sin(x) \\
    g(x) &= \cos(x)
  \end{aligned}, \qquad x \in [0, 1]
\]

\[
  \begin{rcases}
    f(x) = \sin(x) \\
    g(x) = \cos(x)
  \end{rcases} \quad x \in [0, 1]
\]

\end{document}

Another possibility using the empheq package

\documentclass{article}

\usepackage{empheq}

\begin{document}


\begin{empheq}[right={\empheqrbrace, \qquad x \in [0, 1]}]{align*}
    f(x) &= \sin(x) \\
    g(x) &= \cos(x)
\end{empheq}


\begin{empheq}[right={, \qquad x \in [0, 1]}]{align*}
    f(x) &= \sin(x) \\
    g(x) &= \cos(x)
\end{empheq}


\end{document}

A last solution with the \ArrowBetweenLines command from mathtools:

\documentclass{article}
\usepackage{mathtools}

\begin{document}

 \begin{alignat*}{2}
    f(x) &= \sin(x) & & \\
\ArrowBetweenLines*[,\enspace \forall x\in {[0,1]}]
    g(x) &= \cos(x) & &
    \end{alignat*}

\end{document}