The multiplication of two independent univariate Gaussian pdfs of variables $x$ and $y$ is equivalent to a multivariate Gaussian of the two variables (with diagonal covariance matrix).
Does the mixture of two such distributions, e.g.
$$ f(x,y) = \alpha \mathcal{N}(x;\mu_1, \sigma_1^2) + \beta \mathcal{N}(y;\mu_2, \sigma_2^2), $$
also form a valid multivariate distribution (probably multimodal)? How can this be interpreted, or is this totally nonsensical?
Makes no sense because:$$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\alpha u(x)+\beta v(y)dxdy$$ cannot take value $1$ if $u$ and $v$ are PDF's.
E.g. have a look at term $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\alpha u(x)dxdy=\int_{-\infty}^{\infty}\alpha dy\in\{0,+\infty,-\infty\}$