Reading the question it seems to me that you are not looking for actual boxed boxes but just want to have text side by side while allowing for page breaks. paracol
supports such features.
\documentclass{article}
\usepackage{paracol}
\usepackage{lipsum}
\begin{document}
\section{A section}
\lipsum[1-4]
\begin{paracol}{2}
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
\switchcolumn[1]
Many more lines of text. Many more lines of text.
Many more lines of text. Many more lines of text.
Many more lines of text. Many more lines of text.
Many more lines of text. Many more lines of text.
Many more lines of text. Many more lines of text.
Many more lines of text. Many more lines of text.
Many more lines of text. Many more lines of text.
Many more lines of text. Many more lines of text.
Many more lines of text. Many more lines of text.
Many more lines of text. Many more lines of text.
Many more lines of text. Many more lines of text.
Many more lines of text. Many more lines of text.
\end{paracol}
\lipsum[4]
\end{document}

You can combine this with tcolorbox
, if needed.
\documentclass{article}
\usepackage{paracol}
\usepackage{lipsum}
\usepackage[breakable]{tcolorbox}
\begin{document}
\section{A section}
\lipsum[1-4]
\begin{paracol}{2}
\begin{tcolorbox}[breakable]
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
\end{tcolorbox}
\switchcolumn[1]
\begin{tcolorbox}[breakable]
Many more lines of text. Many more lines of text.
Many more lines of text. Many more lines of text.
Many more lines of text. Many more lines of text.
Many more lines of text. Many more lines of text.
Many more lines of text. Many more lines of text.
Many more lines of text. Many more lines of text.
Many more lines of text. Many more lines of text.
Many more lines of text. Many more lines of text.
Many more lines of text. Many more lines of text.
Many more lines of text. Many more lines of text.
Many more lines of text. Many more lines of text.
Many more lines of text. Many more lines of text.
\end{tcolorbox}
\end{paracol}
\lipsum[4]
\end{document}

If you want to keep these boxes, you probably want to change their layout.
As you say that they should be invisible, here is a way to achieve this.
\documentclass{article}
\usepackage{paracol}
\usepackage{lipsum}
\usepackage[breakable]{tcolorbox}
\newtcolorbox{invbox}[1][]{breakable,boxrule=0pt,colback=white,colframe=white,#1}
\begin{document}
\section{A section}
\lipsum[1-4]
\begin{paracol}{2}
\begin{invbox}
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
Many lines of text. Many lines of text.
\end{invbox}
\switchcolumn[1]
\begin{invbox}
Many more lines of text. Many more lines of text.
Many more lines of text. Many more lines of text.
Many more lines of text. Many more lines of text.
Many more lines of text. Many more lines of text.
Many more lines of text. Many more lines of text.
Many more lines of text. Many more lines of text.
Many more lines of text. Many more lines of text.
Many more lines of text. Many more lines of text.
Many more lines of text. Many more lines of text.
Many more lines of text. Many more lines of text.
Many more lines of text. Many more lines of text.
Many more lines of text. Many more lines of text.
\end{invbox}
\end{paracol}
\lipsum[4]
\end{document}

However, in this case I'd probably just go with the plain paracol
solution and change some indents if needed.