There are two separate issues related to the incorrect box size.
First, the textstyle parameter is not used when the label is put inside of a box for measuring the size. This can be considered a bug. It can be patched by adding the parameter to the contents of the box, similar to the code used for actually typesetting the label (which is separate from the code used to measure the dimensions).
\xpatchcmd{\dochronoevent}{%
% search
\ifdim\!chrperiodeventtextwidth > \z@%
\vbox {\hsize\!chrperiodeventtextwidth #3}%
\else #3\fi%
}{%
% replace
\ifdim\!chrperiodeventtextwidth > \z@%
\vbox {\hsize\!chrperiodeventtextwidth \!chr@eventtextstyle{#3}}%
\else \!chr@eventtextstyle{#3}\fi%
}{}{}
The second issue is the date=false parameter. This part of the box is actually measured separately, and the box is smaller when the parameter is false, however for some reason the measured reduction in size is not large enough to actually fully compensate for the date label. I'm not really sure why that happens. A solution for this is to subtract a bit of extra space when the parameter is false - it is not very clean but it seems to work.
\xpatchcmd{\dochronoevent}{%
\if!chreventdate\hbox{#2}\fi}{%
\if!chreventdate\hbox{#2}\else\vskip-6pt\fi}{}{}
Note that chronosys uses the ! character in macro names, so apart from \makeatletter you also need to change the catcode of !.
Full MWE:
\documentclass{article}
\usepackage{chronosys}
\usepackage{tcolorbox}
\usepackage{xpatch}
\makeatletter
\catcode`\!=11
% take textstyle into account when measuring size
\xpatchcmd{\dochronoevent}{%
\ifdim\!chrperiodeventtextwidth > \z@%
\vbox {\hsize\!chrperiodeventtextwidth #3}%
\else #3\fi%
}{%
\ifdim\!chrperiodeventtextwidth > \z@%
\vbox {\hsize\!chrperiodeventtextwidth \!chr@eventtextstyle{#3}}%
\else \!chr@eventtextstyle{#3}\fi%
}
{\typeout{text style patch ok}}
{\typeout{text style patch failed}}
% remove 6pt when date is not printed
\xpatchcmd{\dochronoevent}{%
\if!chreventdate\hbox{#2}\fi}{%
\if!chreventdate\hbox{#2}\else\vskip-6pt\fi}
{\typeout{date correction patch ok}}
{\typeout{date correction patch failed}}
\catcode`\!=12
\makeatother
\begin{document}
\noindent
\begin{minipage}{0.45\linewidth}
\begin{tcolorbox}
\startchronology[startyear=1, stopyear=2000, dates=false]
\chronoevent{1500}{\parbox{2cm}{%
\centering Some text that goes on to a few lines by way of example.}}
\stopchronology
\end{tcolorbox}
\end{minipage}\hfill
\begin{minipage}{0.45\linewidth}
\begin{tcolorbox}
\startchronology[startyear=1, stopyear=2000, dates=false]
\chronoevent[date=false, textstyle=\footnotesize]{1500}{\parbox{2cm}{%
\centering Some text that goes on to a few lines by way of example.}}
\stopchronology
\end{tcolorbox}
\end{minipage}
\end{document}
Result:
