Multivariable Taylor expansion does not work as expected

It's true that the multivariable version of Series can't be used for your purpose, but it's still pretty straightforward to get the desired order by introducing a dummy variable t as follows:

Normal[Series[f[(x - x0) t + x0, (y - y0) t + y0], {t, 0, 2}]] /. t -> 1

$(x-\text{x0}) (y-\text{y0}) f^{(1,1)}(\text{x0},\text{y0})+\frac{1}{2} (x-\text{x0})^2 f^{(2,0)}(\text{x0},\text{y0})+(x-\text{x0}) f^{(1,0)}(\text{x0},\text{y0})+(y-\text{y0}) f^{(0,1)}(\text{x0},\text{y0})+\frac{1}{2} (y-\text{y0})^2 f^{(0,2)}(\text{x0},\text{y0})+f(\text{x0},\text{y 0})$

The expansion is done only with respect to t which is then set to 1 at the end. This guarantees that you'll get exactly the terms up to the total order (2 in this example) that you specify.

Here's an attempt:

multiTaylor[f_, {vars_?VectorQ, pt_?VectorQ, n_Integer?NonNegative}] :=
                   Sum[Nest[(vars - pt).# &, (D[f, {vars, \[FormalK]}]
                       /. Thread[vars -> pt]), \[FormalK]]/\[FormalK]!,
                       {\[FormalK], 0, n}, Method -> "Procedural"]

multiTaylor[f[x, y], {{x, y}, {0, 0}, 2}]
   f[0, 0] + y*Derivative[0, 1][f][0, 0] + x*Derivative[1, 0][f][0, 0] +
   (y*(y*Derivative[0, 2][f][0, 0] + x*Derivative[1, 1][f][0, 0]) +
    x*(y*Derivative[1, 1][f][0, 0] + x*Derivative[2, 0][f][0, 0]))/2

Jens answered this nicely. But my preferred way of thinking about this question is via re-scaling (essentially using perturbation theory) instead of dummy variables.

For example, to generate the series expansion, re-scale all variables by s and expand by series coercion:

(f[x,y] /. Thread[{x,y} -> s {x,y}]) + O[s]^3

This form is generally useful—and although one can use Normal, often working with the series is what you really end up wanting to do.