Can a polynomial of degree 2 be non quadratic By wikipedia:
In algebra, a quadratic function, a quadratic polynomial, a polynomial of degree 2, or simply a quadratic, is a polynomial function in one or more variables in which the highest-degree term is of the second degree.

But the below polynomial is also of degree 2 and non quadratic:

xy + x + y

So, why do we say, polynomial of degree 2 is a quadratic function?
We say it because of conventions. It makes sense.

I think the biggest reason why is that whether a function on the plane is quadratic shouldn't depend on exactly which coordinate axes we impose on the plane. For instance, if I didn't like your $xy$-axes, and instead wanted $uv$-axes $45^\circ$ to yours (say my point $(1,0)$ is at your $(1,1)$ and my point $(0,1)$ is at your $(-1,1)$), then my expression for the same function would be
$$
\frac14u^2-\frac14v^2 + u
$$
which clearly has quadratic terms.
$xy+x+y =0$ (I added the = 0 to make your question complete) is not of degree two for x.

Rearranging: $y=\frac{-x}{x+1}$

Thus it is not quadratic.