Without loss of generality, let an ellipse be centered on the origin (0,0) with the major axis aligned with the 45 degree line (y=x).
Given the lengths of the major and minor axes, the ellipse is uniquely specified. But if the only information given are the lengths of two arbitrarily-oriented perpendicular axes, and the angle between one of the said axis and the major axis, can be the ellipse be specified uniquely?
For example, if it is known the ellipse intersects the x-axis at 5, and intersects the y-axis at 5, can be ellipse be uniquely specified? (Remember the ellipse is centered at (0,0) and the major axis is along y=x).
No, the ellipse as described is not uniquely specified.
For your example, any ellipse in the family $$ \frac{(x-y)^2}{a^2} - \frac{(x+y)^2}{b^2} =1 $$ with $$\frac{1}{a^2} +\frac{1}{b^2} = \frac{1}{5^2} $$ will pass thru $(5,0)$ and $(0,5)$ One of these is a nice circle of radius $5$, another (with $a = \frac{25}{3}, b=\frac{25}{4}$) is an ellipse with axes in ratio 4:3.