Each dimension represents a feature/aspect of the collected data and
has its own Gaussian distribution.
Not exactly. When we speak of "a mixture of gaussians" each gaussian is a "component" of that mixture. But each one of those gaussians might be one-dimensional (only one feature) or multi-dimensional (several features). In the later case, we have for each component a single multidimensional gaussian variable which has a joint gaussian distribution: don't think each feature by itself is a gaussian variable (it is, but that's not enough).
For example, suppose we measure the weight and height (w,h) of the men and women who live in some town. Suppose that, for each population (men and women), $(w,h)$ follows a gaussian distribution. Then, for each population, the observed variable is a two-dimensional gaussian variable (two features). Now, the weight-height of all the people (mixture of men and women) will be the mixture of those (two dimensional) gaussians. Hence, we have here two dimensions (weight-height) and two components (men and women).