$\because$正方形$ABCD$与正方形$EFGH$是位似图形，点$M$的坐标为$\left(-1,1.5\right)$，点$F$的坐标为$\left(2,0\right)$，
$\therefore$正方形$ABCD$与正方形$EFGH$的相似比为$1:3$，
$\because$点$A$，$B$的坐标为$\left(0,2\right)$，$\left(0,1\right)$，
$\therefore AB=1$，
$\therefore EF=3$，
$\because$四边形$EFGH$为正方形，
$\therefore FH=EF=3$，
$\therefore H$的坐标为$\left(5,3\right)$.
故答案为：$\left(5,3\right)$.

$\because$正方形$ABCD$与正方形$EFGH$是位似图形，点$M$的坐标为$\left(-1,1.5\right)$，点$F$的坐标为$\left(2,0\right)$，
$\therefore$正方形$ABCD$与正方形$EFGH$的相似比为$1:3$，
$\because$点$A$，$B$的坐标为$\left(0,2\right)$，$\left(0,1\right)$，
$\therefore AB=1$，
$\therefore EF=3$，
$\because$四边形$EFGH$为正方形，
$\therefore FH=EF=3$，
$\therefore H$的坐标为$\left(5,3\right)$.
故答案为：$\left(5,3\right)$.