Somebody back me up---there are all kinds of Google hits for this particular phrase and the word "generically" itself, and I think it's being used incorrectly to mean "generally."
As your dictionary no doubt told you, "generic" is derived from the Latin "genus" meaning class or type. It means "pertaining or appropriate to large classes or groups as opposed to specific members of the group." Thus, I might say, "I need a generic white blouse, something I can wear to work with a suit and also with jeans on the weekend."
"Generic" entered into common use in America in, I believe, the 1970s, when in response to wildly inflating costs, groceries began offering "generic label" foods. These were packaged in plain white or yellow labels with names like "COLA" or "GREEN BEANS" in stark capital letters. The idea was that the cost savings from not having to advertise the brand, print a fancy label, etc. were passed on to the consumer. Also, "generic drugs"---drugs manufactured and sold without the brand name after the patent has expired---represent a big savings to patients.
Anyway, my hunch is that gradually, the very similar word "generic," which sounds much grander than "general," has come to replace it in phrases such as the one you quoted. Without context, I would guess that "[generally] generically aware" in the sentence you quoted was meant to mean "without reference to detail." "I was generally aware that Krispy-Kreme donuts were unhealthy, but I had no idea they had twelve grams of fat apiece."
Anybody else want to jump in?
Generic brand - Wikipedia, the free encyclopedia