You can use \pgfgetlastxy or \pgfextractx and \pgfextracty. One uses macros and the other uses length registers.
\documentclass{standalone}
\usepackage{tikz}
\usepackage{twemojis}
\newlength{\mywidth}
\newlength{\myheight}
\newcommand{\getwh}[1]{% #1 = node name (no parens)
\pgfextractx{\mywidth}{\pgfpointdiff{\pgfpointanchor{#1}{west}}{\pgfpointanchor{#1}{east}}}%
\pgfextracty{\myheight}{\pgfpointdiff{\pgfpointanchor{#1}{south}}{\pgfpointanchor{#1}{north}}}%
}
\begin{document}
\begin{tikzpicture}
\tikzset{nodestyle/.style={rectangle, rounded corners, draw=black, thick, fill=black!50}}
\node[nodestyle] (node00) at (0, 0) {{\LARGE\texttwemoji{dollar banknote}}};
\getwh{node00}% get size based on anchors
\addtolength{\mywidth}{-0.8pt}% remove line thickness
\addtolength{\myheight}{-0.8pt}% remove line thickness
\node[nodestyle, right=0pt, minimum width=\mywidth, minimum height=\myheight] at (node00.east) (node01) {};
\end{tikzpicture}
\end{document}