Is there a name for the literary device by which two parts or aspects of the same whole are referred to as separate wholes? For example:
There is only one morning in the first example, and one “face” in the second example, but we talk as if there were two of each.
Does this device have a specific name? And are there prominent examples that illustrate its use?
The term is Synecdoche (pronounced “sin-ek-doe-kee”, not “sin-ek-doshe”, incidentally. 
). Examples in the Wikipedia article linked above.
The figure of speech that I am describing is an example of synecdoche. But synecdoche usually involves a part standing for the whole. I am wondering if there is a more specific term when different parts or aspects of the same whole are referred to as if they were each an independent whole.
Ahh, I see what you mean.
I don’t think there’s a more specific term than synecdoche for this particular device, although the two examples you give can also be described as syncrisis. “syncritical synecdoche”, perhaps?
My understanding has always been that “synecdoche” refers to either the part standing for the whole (“I’m gonna get some wheels”) or the whole standing for the part (“Washington announced today…”).
It is not the same thing you are talking about, but when a poem is written about something by describing each of its parts in turn (especially if the thing is a woman, and the poem is affectionate) it is called a blazon.
Yes, the OP’s examples are synecdoche; in the first one, the weather in Boston and New York are parts of the whole “morning”, in the second, “face” and “gaze” are parts of the whole demeanour/character of the person described.
The question is: is there a specific name for the figure of speech when two seperate synecdochies (synechdochia? Don’t know Greek grammar that well, I’m afraid) are used to contrast two aspects of the whole they refer to?
Incidentally, there’s a very good example of it, whatever it’s called, in “Hamlet”, Scene 2, Lines 12-14:
Have we, as 'twere with a defeated joy,–
With an auspicious and a dropping eye,
With mirth in funeral and with dirge in marriage,
In equal scale weighing delight and dole,–
Taken to wife: