Remove space after \captionof

It is sometimes confusing to distinguish between aboveskip and belowskip because their meaning sometimes change according to the value of position. It is discussed in section 2.6 of caption's documentation. To avoid confusion in a case like this, it might be better to set skip to zero.

\documentclass{article}
\usepackage[svgnames,usenames,dvipsnames,table]{xcolor}
\usepackage[showframe=false]{geometry}
\usepackage[many]{tcolorbox}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{caption}

\usepackage{lipsum}

\newtcolorbox{texample}[1][]{float, notitle, #1}

\usepackage{float}
\floatstyle{plain}
\newfloat{example}{thp}{lop}
\floatname{example}{\textsc{Figure}}


\begin{document}
\lipsum[2]

\begin{texample}[bicolor,sharp corners,
    colback=white,colframe=gray!20,colbacklower=gray!20,
    top=1mm,bottom=1mm,middle=1mm]
\centering
\includegraphics[width=.25\linewidth]{example-image}
\tcblower
\captionsetup{skip=0pt}
\captionof{example}{Caption content}
\end{texample}

\lipsum[2]

\end{document}