And that Cullen friendship tile is somewhat perplexing as well because at no point do I recall there being an option for Hawke to become overtly friendly or unfriendly with him. There were options where you could agree or disagree with his views, but that was about it. Is that what is being used as the basis for their friendship? I doubt it because that probably would (or should) have raised a conflict flag for anyone that wanted to be friendly with him but sided with the Mages overall.
This is pretty arcane and few people seem to have noticed it, but Cullen does change his disposition towards Hawke in Act 3 depending on how you've spoken to him and/or how you've been handling the mage/templar-related quests. If you're very pro-mage, to the point of openly arguing with him about it (with such dialogue options as "It's the Templars' fault" / "mages must be free!"), in Act 3 Cullen will greet Hawke with "you've been less of a friend to the Templars." If you've been neutral/agreeable/not argumentative in dialogue with him, he will greet you as "good to see an old friend."
It does take you at least two radically different playthroughs paying special attention to a rather minor character to notice that this is even there at all, but it is. Generally, the more neutral/pro-Templar you are, the more friendly he becomes, and the more abrasive/pro-mage you are, the less friendly he gets. It is possible to play pro-mage and still be friendly with Cullen, though - you just can't be excessively contrarian when talking to him. If you go for neutral dialogue choices (say, rather than "all mages must be free" or "all Templars are evil," you answer "both sides have their flaws" or something), he'll still regard you as a friend.
For Keep purposes, because this would take an unwieldy amount of work to go back through your saves to check (if you even have them), this choice can be made based on your Hawke's personality/beliefs. You can guess whether your Hawke is the type to make friends with Templars or not from their general disposition. And ultimately, it can be treated as more of "how do you want Cullen to remember Hawke in DAI" question. Even if this flag turns out to not do anything after all, it's a good way to guess what you should choose for each of your Hawkes.