E.g. adding text depth=0.25ex,text height=0.8em yields
\documentclass[border=1mm]{standalone}
\usepackage{tikz}
\usetikzlibrary{matrix}
\def\h{0.55cm}
\def\w{1.5cm}
\tikzset{
myboxshape/.style={rectangle, outer sep=0, minimum width = \w, minimum height = \h, inner sep=0},
mybox/.style={myboxshape, very thin, draw,text depth=0.25ex,text height=0.8em},
}
\begin{document}
\begin{tikzpicture}
\matrix
{
\node[mybox] {Vertical}; & \node[mybox] {grungy}; \\
};
\end{tikzpicture}
\end{document}

Notice that there are many ways to simplify this. They include the use of matrix of nodes, i.e. you load the matrix library but do not really use it.
\documentclass[border=1mm]{standalone}
\usepackage{tikz}
\usetikzlibrary{matrix}
\begin{document}
\begin{tikzpicture}[myboxshape/.style={rectangle, outer sep=0,
minimum width = \pgfkeysvalueof{/tikz/mydims/w},
minimum height = \pgfkeysvalueof{/tikz/mydims/h}, inner sep=0},
mybox/.style={myboxshape, very thin, draw,text depth=0.25ex,text height=0.8em},
mydims/.cd,h/.initial=0.55cm,w/.initial=1.5cm]
\matrix[matrix of nodes, nodes={mybox}]
{
Vertical & grungy \\
};
\end{tikzpicture}
\end{document}
text depth=0.25ex, say. – Feb 21 '20 at 19:35