The problem is caused by the \topskip glue, \fboxsep and \fboxrule parameters. Here is an example that works, with exaggerated \fboxsep and \fboxrule values for illustration purposes:
\documentclass[10pt]{article}
\usepackage{xcolor}
\usepackage{calc}
\usepackage[landscape]{geometry}
\geometry{top=0cm, bottom=0cm, left=0cm, right=0cm}
\definecolor{sidebar}{RGB}{71,62,136}
\setlength{\parindent}{0pt}
\setlength{\fboxsep}{10pt}
\setlength{\fboxrule}{20pt}
\begin{document}
\setlength{\topskip}{0pt}
\fcolorbox{black}{sidebar}%
{%
\begin{minipage}[t][\textheight-2\fboxsep-2\fboxrule][t]{0.2\textwidth}
\color{white} % White text
\LARGE Hello
\end{minipage}%
}
\end{document}

The gray color in this screenshot is from the ”behind-page background” displayed by my PDF viewer; it doesn't belong to the page but shows that the \fcolorbox completely fills the page.
Here is a similar example with normal \fboxsep and \fboxrule values, and two identical pages in order to illustrate how to set and restore \topskip to its default value (see below):
\documentclass[10pt]{article}
\usepackage{xcolor}
\usepackage{calc}
\usepackage[landscape]{geometry}
\geometry{top=0cm, bottom=0cm, left=0cm, right=0cm}
\definecolor{sidebar}{RGB}{71,62,136}
\setlength{\parindent}{0pt}
\begin{document}
%\bgroup
\setlength{\topskip}{0pt}
\fcolorbox{sidebar}{sidebar}%
{%
\begin{minipage}[t][\textheight-2\fboxsep-2\fboxrule][t]{0.2\textwidth}
\color{white} % White text
\LARGE Hello
\end{minipage}%
}
\newpage
%\egroup
\fcolorbox{sidebar}{sidebar}%
{%
\begin{minipage}[t][\textheight-2\fboxsep-2\fboxrule][t]{0.2\textwidth}
\color{white} % White text
\LARGE Hello
\end{minipage}%
}
\end{document}
Uncomment the \bgroup and \egroup calls to restore \topskip to its normal value after the first page has been shipped out; then you'll see
Overfull \vbox (1.6pt too high) has occurred while \output is active [2]
for the second page, as expected. The following screenshot was taken in this situation; the first page has \topskip=0pt, while on the second page, it has its default value of 10pt (which can be seen with \showthe\topskip after the uncommented \egroup command):

Notes:
\topskip is glue added between the top of the page (according to the margins) and the baseline of first box on a page. The height of the first box is withdrawn from \topskip to compute the glue automatically inserted by TeX; moreover, the inserted glue is set to zero in case the result would be negative—i.e., when the first box is higher than \topskip.
Your example is missing xcolor.
\paperheightin height) or to be laid on top of it (e.g., centered within the sidebar). – frougon May 27 '22 at 09:17