Sometimes I see the definition of an ODE as a function of two variables $f(x,y)$, e.g. $$ \frac{dy}{dx}=f(x,y) $$ Does it mean $y=y(x,y)$?
Isn't an ODE always a function of one variable?
For instance here they use $y'=f(t,y)$. How should I interpt it?
No, it doesn't. As pointed out we have for example the ODE $y'(x)=\cos(x)+y(x)$. That means on the right hand side we have some dependency on $x$ and $y(x)$. Therefore we write shortly $$y'=f(x,y)$$ where the first component indicates the variable of your ODE and in the second component we have the function $y$ that dependents on $x$. There are also ODEs of higher order, for example $y''(x)=y'(x)+y(x)+4x^2$. Then we write shortly $$y''=f(x,y,y')$$ where in the example we have $f(x,y,y')=y'+y+4x^2$. I think you get the idea. Generally for an $n$-th order ODE $$y^{(n)}=f(x,y,y',y'',...,y^{(n-1)}).$$
Firstly an ODE is no function. It is a special type of equation I'd say. And as seen it also depends on the solution $y$ and its derivatives. But in some way you are right - the solution depends solely on one variable. If it depends on more variable we are in the setting of partial differential equations.
Don't be irritated by this. Now we just have $t$ as the variable. For example $$y'(t)=t^3+\sin(y(t)) \rightarrow y'=f(t,y)$$ where $f(t,y)=t^3+\sin(y)$. We can freely choose the variables, have a look at the differential equations
$$\begin{align} g'(h)=h^2+\ln(g(h)) &\rightarrow g'=f(h,g) \\ \theta''(\zeta)=e^{\zeta}+\theta(\zeta)+2\theta'(\zeta)&\rightarrow \theta''=f(\zeta,\theta,\theta') \\ \kappa'(\mathcal{O})=4\mathcal{O}+\kappa(\mathcal{O})^2 &\rightarrow \kappa'=f(\mathcal{O}, \kappa)\end{align}$$
An example of a differential equation: $$ \frac{dy}{dx} = x^2+y^2 $$ Of course $y$ is supposed to be a function of $x$ only. In your general formulation, I took $f(x,y) = x^2+y^2$. But the point of that definition is that if you choose any function $f$ of two variables, then you will get an ODE.
note Your definition is only for an ODE of order $1$. An ODE of order $2$ will be something like $$ y''= g(x, y, y') $$ Still, $y$ is a function of only one variable $x$, But now $g$ is a function of three variables.
If $f$ is a variable of $2$ functions, it doesn't mean that those variables can't be dependent of one-another. In this case $f(x,y)=f(x,y(x))$. In other words, you can replace $f(x,y)$ with variable $x$ and a function $y$. For example, if we look at an ODE $$y'+y+x=0.$$ We can write it as $$y'=-y-x\Rightarrow y'=f(x,y),$$ where $f(x,y)=-y-x$.