I think that there are two main kinds of "apologies to" acknowledgments connected to instances of paraphrasing or word or phrase borrowing. One is the sincere "I have appropriated this word or expression from person X, who used it in a particular way, and I may not be doing justice to that person's original intention" kind—an acknowledgment that really has an apologetic element to it. For example, from J.J. Godfrey, A Philosophy of Human Hope (1987):
Work without hope draws nectar in a sieve,/And hope without an object cannot live. —Coleridge, Work Without Hope
Our reflection on hope that has an aim or target begins with the features of such targets. With apologies to Coleridge, however, hope does not, strictly speaking, have an "object." What is hoped for is not a thing or an item, but rather a state of affairs or an event.
The other is the utterly insincere "I just invented a clever burlesque of someone else's famous phrase, but I'm not sure you'll recognize it unless I identify the source that I corrupted" kind. These are, I'm sorry to say, quite common. For example, from Robert Horn, How Will They Know If I'm Dead? Transcending Disability and Terminal Illness (1996):
He once wandered through the house for 10 minutes with a used bedpan, flexing all the way, apparently wondering, with apologies to MacBeth, "Is this a bedpan I see before me? If so, what the hell do I do with it?" Get a life, dude!
(That's Macbeth, by the way, dude.)
Many instances of "apologies to" fall somewhere between these two pure types of apology, but I think that the majority tend toward the insincere and self-aggrandizing end of the spectrum.