On various sites, jQuery's $ variable is referred to as an alias, and on others, it is referred to as a factory. I took a look at the source code, and I think the former is correct. As far as I can see, the dollar symbol is being defined here:
// Expose jQuery to the global object
return (window.jQuery = window.$ = jQuery);
This is setting both $ and jQuery to the SAME alias; there is no 'factory' for jQuery objects. Are the sites referring to $ as a 'factory' simply wrong?
I think either is an okay term. The dollar sign is certainly an alias for the jQuery function, specifically to be used as a shorthand. If someone refers to the function as a factory, I don't think that's specific to the $ alias, but just describing what the jQuery function does in general, which is create objects from various different types of input.
The jQuery function (on the right of the assignment in your question) is a factory (it creates new jQuery objects). It is not in the global scope, because it is defined in the scope of a self-executing function.
window.$ and window.jQuery are global aliases for the constructor.