Enumerate list with each item in a new row

I propose one of these two solutions:

\documentclass[8pt,a4paper]{extarticle}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{xcolor}
\usepackage{enumitem}
\usepackage{tabularx}
\usepackage[left=1.2cm, right=1.2cm, top=0.0cm, bottom=0.0cm, headheight=37pt, includeheadfoot]{geometry}
\makeatletter
\newcommand*{\compress}{\@minipagetrue}
\makeatother
\usepackage{makecell}
 \renewcommand{\theadfont}{\color{black}\bfseries}
\usepackage{listliketab}

\begin{document}
\sffamily
\setlength{\extrarowheight}{2pt}
\noindent\begin{tabularx}{\textwidth}{>{\compress}X|>{\color{red}}c|c|c|*{2}{>{\color{red}}c|}}
\textbf{Bezeichnung} & \thead{Einzelpreis} & \textbf{Anzahl}
& \textbf{Intervall} & \thead{Rabatt} & \thead{€/Monat}\\
\hline%<----->
\begin{enumerate}[label=\Roman*. ,widest =VI, noitemsep, wide=0pt, leftmargin=*, after =\vspace*{-\dimexpr\topsep + \partopsep}]
    \item System: Software as a Service Geltungsbereich Arbeits- und
    Gesundheitsschutz % &40,00€ & 1&Laufzeit &10\% &36,00€ \\ <--- Idea but doesn´t work...
    \item IT-Service-Pack Release, Update, Support:\newline
    Übernahme der kompletten IT-Administration und weitere Dienstleistungen
    \item Management: Auswahl-, Bestellung-, Überwachungs- und
    \item Audit zur Evaluation und Qualitätssicherung %& 7,39€ & 1 & Laufzeit & 10\% & 6,46€ \\ <--- I need smth like this...
\end{enumerate} &40,00€ & 1&Laufzeit &10\% &36,00€ \\ %<---- Only this works
\hline%<----->
Summe & & & & &36,46€ 
\end{tabularx}
\vskip1cm

\storestyleof{enumerate}
\begin{listliketab}
\newcounter{tabenum}\setcounter{tabenum}{0}
\newcommand{\nextnum}{\addtocounter{tabenum}{1}\Roman{tabenum}.}
\noindent\begin{tabularx}{\textwidth}{LX|>{\color{red}}c|c|c|*{2}{>{\color{red}}c|}}
    \multicolumn{2}{l|}{\textbf{Bezeichnung}} & \thead{Einzelpreis} & \textbf{Anzahl}
    & \textbf{Intervall} & \thead{Rabatt} & \thead{€/Monat}\\
    \hline\noalign{\vskip-1.8ex}%<----->
\nextnum & System: Software as a Service Geltungsbereich Arbeits- und
        Gesundheitsschutz &40,00€ & 1&Laufzeit &10\% &36,00€ \\[-3ex]
\nextnum & IT-Service-Pack Release, Update, Support:\newline
        Übernahme der kompletten IT-Administration und weitere Dienstleistungen & & & & & \\[-3ex]
\nextnum &Management: Auswahl-, Bestellung-, Überwachungs- und & & & & & \\[-3ex]
\nextnum & Audit zur Evaluation und Qualitätssicherung & 7,39€ & 1 & Laufzeit & 10\% & 6,46€ \\
\hline
\multicolumn{2}{l|}{Summe} & & & & &36,46€
\end{tabularx}
\end{listliketab}

\end{document} 

Similarly to @Bernard answer, but some details are different:

• for the first column is defined column type E for enumerated contents:

\makeatletter
\newcommand*{\compress}{\@minipagetrue}
\makeatother
\newcolumntype{E}[1]{ >{\compress\enumerate}p{#1}<{\endenumerate}}

which simplified writing lists in the first column

• for enumerate list is defined the following setup:

\setlist[enumerate]{%
                 resume=column,
                 nosep,
                 label=\Roman*.,
                 leftmargin=*,
                 after=\vspace*{-\dimexpr\baselineskip + \partopsep}}

• for resuming enumeration of list is used option resume=column, in enumerate list set up

• for start enumeration at I. is dedicated \setcounter{enumi}{0} before first \item in column (see MWE below)

• I didn't bother with coloring numbers (it wasn't clear to me, if in some column all number are red, or they are select according to some other criteria)

• for compilation is used xelatex (due to use € characters for currency)

  Complete MWE is:

\documentclass[8pt,a4paper]{extarticle}
\usepackage[hmargin=1.2cm, vmargin=0.0cm, 
            headheight=37pt, includeheadfoot]{geometry}
\usepackage{enumitem}
\usepackage{ragged2e}
\usepackage{tabularx}

\makeatletter
\newcommand*{\compress}{\@minipagetrue}
\makeatother
\newcolumntype{E}[1]{ >{\compress\enumerate}p{#1}<{\endenumerate}}
\newcommand*\mcbf[1]{\multicolumn{1}{>{\bfseries}l|}{#1}}

\begin{document}

\setlist[enumerate]{  label=\Roman*.,
                 nosep,
                 leftmargin=*,
                 after=\vspace*{-\dimexpr\baselineskip + \partopsep}}
\begin{tabularx}{\linewidth}{ >{\RaggedRight}E{5cm}*{5}{|X}|}
\mcbf{Bezeichnung}    
    & \mcbf{Einzelpreis}    & \mcbf{Anzahl}   & \mcbf{Intervall} 
    & \mcbf{Rabatt}         & \mcbf{€/Monat}\\
    \hline%<----->
    \setcounter{enumi}{0}
\item   System: Software as a Service  Geltungsbereich Arbeits- und Gesundheitsschutz
    & 40,00€    & 1     & Laufzeit  & 10\%  & 36,00€    \\ 
\item   IT-Service-Pack Release, Update, Support: 
    
        Übernahme der kompletten IT-Administration und weitere Dienstleistungen
\item Management: Auswahl-, Bestellung-, Überwachungs- und
\item Audit zur Evaluation und Qualitätssicherung 
    & 7,39€     & 1     & Laufzeit  & 10\%  & 6,46€     \\ 
    \hline%<----->
\mcbf{Summe}
    &           &       &           &       & 36,46€    \\
\end{tabularx}

\end{document}

addendum: initiated by conversation with @Bernard (in comments below) see if the following variation of mine original answer is more appropriate for you:

• definition of the enumerate style in tables is moved in preamble, where is now available to all lists in table
• for the first column is used X column type
• for columns with numbers are used S columns defined in the siunitx package
• your table is encapsulated in table floating environment.
• added are red colors for the second, fifth and six (last) column

\documentclass[8pt,a4paper]{extarticle}
\usepackage[hmargin=1.2cm, vmargin=0.0cm,
            headheight=37pt, includeheadfoot]{geometry}
\usepackage{enumitem}
\usepackage{etoolbox}      % new 
\AtBeginEnvironment{table}{% for enumerate list in "table" environments
\setlist[enumerate]{%      % moved in preamble 
                 resume=column,
                 nosep,
                 label=\Roman*.,
                 leftmargin=*,
                 after=\vspace*{-\dimexpr\baselineskip + \partopsep}}
                            }
\usepackage{ragged2e}
\usepackage{tabularx}
\usepackage{xcolor}
\usepackage{siunitx}

\makeatletter
\newcommand*{\compress}{\@minipagetrue}
\makeatother
\newcolumntype{E}{ >{\compress\enumerate}X<{\endenumerate}}
\newcommand*\mcbf[1]{\multicolumn{1}{>{\bfseries}l|}{#1}}

\begin{document}
    \begin{table}
\begin{tabularx}{\linewidth}{@{}
            >{\RaggedRight}E
            |S[table-format=2.2,color=red]<{\,\textcolor{red}{€}}
            |S[table-format=1.0]
            |l
            |S[table-format=2.0,color=red]<{\,\textcolor{red}{\%}}
            |S[table-format=2.2,color=red]<{\,\textcolor{red}{€}}|
                            }
\mcbf{Bezeichnung}
    & \mcbf{Einzelpreis}    & \mcbf{Anzahl}   & \mcbf{Intervall}
    & \mcbf{Rabatt}         & \mcbf{€/Monat}\\
    \hline
    \setcounter{enumi}{0}       % <--- reset counter enumi
\item   System: Software as a Service  Geltungsbereich Arbeits- und Gesundheitsschutz
    & 40,00     & 1     & Laufzeit  & 10            & 36,00 \cr % <---
    \setcounter{enumi}{1}
\item   IT-Service-Pack Release, Update, Support:

        Übernahme der kompletten IT-Administration und weitere Dienstleistungen
\item Management: Auswahl-, Bestellung-, Überwachungs- und
\item Audit zur Evaluation und Qualitätssicherung
    & 7,39      & 1     & Laufzeit  & 10            & 6,46 \cr % <---
    \hline%<----->
\mcbf{Sume}
    & \mcbf{}   &       &           & \mcbf{}       & 36,46 \cr % <---\
\end{tabularx}
    \end{table}
\end{document}