What is the relation between the domain of Fractional Fourier transform and time and frequency?
It think it is more correct to say that the $u$ axis is a linear combination of the $x$ and $\xi$ axes:

$$(x, \xi) = (1, 0)x + (0, 1)\xi = \left((1, 0)\cos(\alpha) + (0, 1)\sin(\alpha)\right)u,$$

that is:

$$ x = u\cos(\alpha)\\
\xi = u\sin(\alpha).$$

The definition of the fractional Fourier transform from that Wikipedia page is:


  $$\mathcal{F}_\alpha[f](u) =  \sqrt{1-i\cot(\alpha)} e^{i \pi
\cot(\alpha) u^2}  \int_{-\infty}^\infty  e^{-i2\pi (\csc(\alpha) u x
- \frac{\cot(\alpha)}{2} x^2)} f(x)\, \mathrm{d}x.$$


If you use another variable, $y$, for the integral, and substitute $u = \sqrt{x^2 + \xi^2}$ and $\alpha = \text{atan}(\xi, x)$, then the fractional Fourier transform can be written as:

$$\mathcal{F}[f](x, \xi) = e^{i \pi \frac{\xi}{x} (x^2+\xi^2)} \sqrt{1-\frac{i \xi}{x}}\int^\infty_{-\infty}e^{i \pi \frac{y}{x} \left(\xi y-2 x^{\frac{1}{\xi}} \sqrt{\xi^2+x^2}\right)} f(y)\, \text{d}y.$$