LaTeX throws error on heightof

heighof doesn't work inside tikz, use the functions provided by it. See the section "Mathematical Expressions" in the documentation.

documentclass[14pt,a4paper]{extarticle}
usepackage{calc}
usepackage{tikz}
begin{document}
begin{tikzpicture}
node[minimum height=height("A")]{A};
end{tikzpicture}
end{document}

Addition: minimum height does not describe the height of the node content, in the sense of the height of a TeX box, but the total height:

documentclass{article}
usepackage{tikz}
begin{document}
begin{tikzpicture}
node[draw,inner sep=0,minimum height=10pt] (A) {.};

draw[red,<->] ([xshift=-2pt]A.south west) --++ (0,10pt);
end{tikzpicture}
begin{tikzpicture}
node[draw,minimum height=10pt] (A) {.};

draw[red,<->] ([xshift=-2pt]A.south west) --++ (0,10pt);
end{tikzpicture}

end{document}

The command heightof only works in calc package expressions.

You can use height{"A"}, but this wouldn't take care of the inner sep, so you should add it to the minimum height.

documentclass{article}
usepackage{tikz}

begin{document}

begin{tikzpicture}
node[draw,minimum height=height("A")+2*(pgfkeysvalueof{/pgf/inner ysep})]{A};
end{tikzpicture}
begin{tikzpicture}
node[draw,minimum height=height("A")+2*(pgfkeysvalueof{/pgf/inner ysep})]{x};
end{tikzpicture}

begin{tikzpicture}[inner sep=1pt]
node[draw,minimum height=height("A")+2*(pgfkeysvalueof{/pgf/inner ysep})]{A};
end{tikzpicture}
begin{tikzpicture}[inner sep=1pt]
node[draw,minimum height=height("A")+2*(pgfkeysvalueof{/pgf/inner ysep})]{x};
end{tikzpicture}

end{document}