Clearly some of us are using the word "affiliated" in different senses. Rather than arguing about the meaning of the word, it would be better to focus on the real question "what is the NATURE of the affiliation/relationship/whatever between Google and the ODP?"
The answer is, from our point of view, NONE. We generate data, we say anyone can use it, and we don't even have any official or comprehensive list of the people that use it. There is an ODP category that lists sites we know about -- they may be related to us, but we aren't related to them. None of them pay us money. If any of them died tomorrow, we wouldn't even get sick.
From Google's point of view, the relationship is producer-consumer. We produce a directory, Google consumes it and republishes it.
Google has been asked about the relationship of the ODP and Google search. Their official answer is: The ODP is in no way different from any other site. Its pages' page rank gets calculated exactly like any other pages, its links get treated exactly like any other links. I can't imagine why they would lie. (Invent your own conspiracy theory here, but I'm not likely to buy it.)
I also can't imagine any way in which that answer was less than comprehensive. Oh, sure, Google shows the ODP category and site description when their search returns one of the few pages actually in the ODP. And their engineers may be busily imagining other clever ways to use the ODP data for spam control or whatever. But if so, they haven't even hinted at it to any audience I've been in. (Not that they would have.)
So take that for what it's worth. Get the difference between Google Search and Google Directory firmly fixed in your mind. Whenever you hear a rumor about Google, ask the rumorer whether they're talking about Search results or the Directory. If their eyes glaze over, add them to your list of people handicapped by being technologically clue-deprived.