Geometric intuition behind the point where the indifference curve meets the budget line
No. This is not in general true.
Maximizing the rectangle formed in the described manner would maximize the product $x_1 \times x_2$. In the special case where your utility function takes the form $u(x_1,x_2)=x_1x_2$, utility maximizing implies you pick a point on your budget constraint where the rectangle you described is maximized.
Suppose however, that instead your utility function is given by $v(x_1,x_2)=x_1^{1/3}x_2^{2/3}$ And suppose you are currently maximizing $x_1 \times x_2$ on your budget constraint. Suppose that $p_1=p_2=1$, then maximizing $x_1 \times x_2$ implies $x_1^*=x_2^*$. However maximizing $v$ with these prices gives: $x_1^*=\frac{1}{2}x_2^*$. Thus not maximizing the rectangle subject to your constraint, but still defining a point of tangency between the indifference curve and budget constraint.