I am not sure if I understand your question, but if you wish to align a vertically with b and c, consider
\documentclass[a4paper]{article}
\usepackage{array,tabularx,tikz,ragged2e,siunitx}
\begin{document}
{\renewcommand\tabularxcolumn[1]{m{#1}}
\renewcommand{\arraystretch}{1.3}
\begin{tabularx}{\linewidth}{%
|X|*{3}{>{\Centering}m{2cm}|}}\hline
Entourer la ou les bonne(s) reponse(s)
&\tikz[baseline=(A.base),overlay]
\node[fill=blue!15,minimum height=20pt,minimum width=2cm+2\tabcolsep] (A) {a};
&b&p\\\hline
\end{tabularx}}
\end{document}

UPDATE: OK, I see that @Zarko already gave you what you wanted. For fun, I followed a bit your original approach, and this is where I got:
\documentclass[a4paper]{article}
\usepackage{array,tabularx,tikz,ragged2e,siunitx}
\usetikzlibrary{calc}
\newcommand{\TikzStrut}[1]{\tikz[overlay,remember picture]{\node(#1){\strut};}}
\makeatletter
\newcommand{\RowHeight}{% see e.g. https://tex.stackexchange.com/a/84536/121799
\def\tmp{\dimexpr\arraystretch\ht\strutbox+\arraystretch\ht\strutbox+\arraystretch\dp\@arstrutbox}\relax
\xdef\myrowheight{\the\tmp}\relax
}
\makeatother
\begin{document}
{\renewcommand\tabularxcolumn[1]{m{#1}}
\renewcommand{\arraystretch}{1.3}
\begin{tabularx}{\linewidth}{%
|X|*{3}{>{\Centering}m{2cm}|}}\hline
Entourer la ou les bonne(s) reponse(s)\RowHeight
&\tikz[baseline=(A.base),overlay,remember picture]{
\node[fill=blue!15,minimum height=\myrowheight,minimum width=2cm+2\tabcolsep] (A)
{a};}
&b&p\\\hline
\end{tabularx}}
\end{document}

It does compute the height of the row correctly (I think) but only after I used the fact that there is a row with two lines. And also the vertical alignment is not perfect. I'm just posting that so that it can be completed if needed later.
fill a tabular box with a tikz node. Tell me I to be more specific : fill -- full place is taken, tabular box -- it's clear, and tikz node also. I know that in anglo-saxon tradition there is not much line, and I appriceate that, but in France and in school, people are used to square boxes ;-) – Tarass Feb 13 '18 at 17:12