How do I force caption to align under the image?

Question

How can I force caption to align to the left corner of the image above it and also not be wider than the image in this code example?

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{graphicx}
\usepackage{caption}        % for source
\usepackage{float}          
\usepackage[export]{adjustbox} % valign
\newcommand{\imagesource}[1]{{\scriptsize Source: #1}}
\usepackage[export]{adjustbox}
    
\begin{document}

\begin{figure}[H]
\centering
    \begin{minipage}[t]{.475\textwidth} % [t][][b]
        \begin{tabular}[t]{ @{} r @{} }
        \includegraphics[width = .9\linewidth,valign=t]{example-image} \\
        \imagesource{(EU 2020)}
       \end{tabular}
    \end{minipage}\hfill
    \begin{minipage}[t]{.475\textwidth}
        \begin{tabular}[t]{ @{} r @{} }
        \includegraphics[width = .7\linewidth,valign=t]{example-image} \\
        \imagesource{(IEA 2019)}
        \end{tabular}
    \end{minipage}
     \begin{minipage}[t]{.475\textwidth}
       \caption{This is a caption for the first image.}\label{fig:image1}
     \end{minipage}\hfill
     \begin{minipage}[t]{.475\textwidth}
        \caption{This is a caption for the second image.}\label{fig:image2}
     \end{minipage}
\end{figure}
\end{document}

Answer 1

Like this?

With use of tabular instead of the minipages:

\documentclass{article}
\usepackage{graphicx}
\usepackage{caption}           % for source
\usepackage[export]{adjustbox} % valign
\newcommand{\imagesource}[1]{{\smallskip\hfill\scriptsize Source: #1}}

\begin{document}

\begin{figure}[ht]
\centering
\setkeys{Gin}{width=\linewidth}
    \begin{tabular}{p{0.45\linewidth} p{0.34\linewidth}}
\includegraphics[valign=t]{example-image}

\imagesource{(EU 2020)}
    & 
\includegraphics[valign=t]{example-image}

\imagesource{(IEA 2019)}    \\
\caption{This is a caption for the first image.}
\label{fig:image1}
    &   \caption{This is a caption for the second image.}
        \label{fig:image2}
    \end{tabular}
\end{figure}

\end{document}

Addendum: You may liked alternative solution with use of the copyrightbox:

\documentclass{article}
\usepackage{graphicx}
\usepackage{caption}        % for source
\newcommand{\imagesource}[1]{\hfill\scriptsize Source: #1}
\usepackage{copyrightbox}
\makeatletter
\renewcommand{\CRB@setcopyrightparagraphstyle}{\raggedleft} % new
\makeatother

\begin{document}

\begin{figure}[ht]
\centering
\setkeys{Gin}{width=\linewidth}
    \begin{tabular}{p{0.45\linewidth}p{0.34\linewidth}}
\copyrightbox[b]{\includegraphics{example-image-duck}}{(EU 2020)}
    & 
\copyrightbox[b]{\includegraphics{example-image-duck}}{(IEA 2019)}    
    \\[-3ex]
\caption{This is a caption for the first image.}
\label{fig:image1}
    &   \caption{This is a caption for the second image.}
        \label{fig:image2}
    \end{tabular}
\end{figure}

\end{document}

Answer 2

The following MWE allows to reproduce the output shown in the original question.

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{graphicx}
\usepackage{caption}        % for source
\usepackage{float}          
\usepackage[export]{adjustbox} % valign
\newcommand{\imagesource}[1]{{\scriptsize Source: #1}}
\usepackage[export]{adjustbox}

\usepackage{fullpage} % full wide page, small margins
\begin{document}

\begin{figure}[H]
\centering
    \begin{minipage}[t]{.475\textwidth} % [t][][b]
        \begin{tabular}[t]{ @{} r @{} }
        \includegraphics[width = .9\linewidth,valign=t]{example-image} \\
        \imagesource{(EU 2020)}
       \end{tabular}
    \end{minipage}\hfill
    \begin{minipage}[t]{.475\textwidth}
        \begin{tabular}[t]{ @{} r @{} }
        \includegraphics[width = .7\linewidth,valign=t]{example-image} \\
        \imagesource{(IEA 2019)}
        \end{tabular}
    \end{minipage}
     \begin{minipage}[t]{.475\textwidth}
       \caption{This is a caption for the first image.}\label{fig:image1}
     \end{minipage}\hfill
     \begin{minipage}[t]{.475\textwidth}
        \caption{This is a caption for the second image.}\label{fig:image2}
     \end{minipage}
\end{figure}
\end{document}

As already speculated in the comments to the previous question, this behaviour is due to the caption package that automatically horizontally centers captions that only span one line. You can turn off this behaviour globally (in the preamble) or locally (in the corresponding figure environment) by using \captionsetup{singlelinecheck=false}. This will result in the expected output: