Your minipage of width \textwidth starts a new paragraph and therefore is indented by the amount of \parindent (15pt in article), so it spills over into the right margin by that amount. An easy fix is to add \noindent before the minipage.
\documentclass[a4paper, 10pt, notitlepage]{article}
\begin{document}
\noindent \verb|\parindent|: \the\parindent
\noindent aaaaaaaaaaaaaaaaaaaaaaa
\begin{minipage}[t]{1.0 \textwidth}
aaaaaaaaaaaaaaaaaaaaaaa
\end{minipage}
\noindent\begin{minipage}[t]{1.0 \textwidth}
aaaaaaaaaaaaaaaaaaaaaaa
\end{minipage}
\end{document}
