You should pass the American english (USenglish) and numeric format options to isodate. This yields the output format mm/dd/yyyy:
01/21/2016
\documentclass{article}
\usepackage[USenglish,num]{isodate}
\begin{document}
\today
\end{document}
The same output under datetime2 (without defining your own date style):
\documentclass{article}
\usepackage[datesep={/}]{datetime2}
\DTMsetdatestyle{mmddyyyy}
\begin{document}
\today
\end{document}
Without any packages, you can just manage the printing yourself using \day, \month (in two-digit format) and \year explicitly:
\documentclass{article}
\makeatletter
\renewcommand{\today}{\two@digits{\number\month}/\two@digits{\number\day}/\number\year}
\makeatother
\begin{document}
\today
\end{document}
For calculations (like a two-digit year) on the date you can use the following:
01/21/16
\documentclass{article}
\usepackage{xparse}
\ExplSyntaxOn
\cs_new_eq:NN \calc \fp_eval:n
\ExplSyntaxOff
\makeatletter
\renewcommand{\today}{%
\two@digits{\number\month}/% MM
\two@digits{\number\day}/% DD
\calc{\number\year-100*trunc(\number\year/100,0)}}% YY
\makeatother
\begin{document}
\today
\end{document}