I have exactly two pairings with Chilton that involve other canon characters, and both of them are abusive, or at least end up like that.
One of them is Childeon, of course. I always imagine them having a history together, knowing each other since med school, and having been in a rather long relationship. That relationship actually started out well, they were happy - the only time Frederick was ever genuinely happy with someone - but at some point, Abel started turning darker and darker, and they eventually broke up.
Today, all that fondness is gone. Frederick himself destroyed the last bits that were left of “his” old Abel, and Abel is just driven by vengefulness. He doesn’t want to see him dead, death would be relief, and Frederick doesn’t deserve relief. He wants to humiliate him, and finally destroy him in return, by any means necessary, no matter how cruel and inhumane. Neither of the two loves the other anymore, but they both very well remember the old times, sometimes with a drop of nostalgia, and a feeling of “How could this happen to us?”, mostly on Frederick’s side. Abel knows that, and he’s absolutely willing to use those feelings against him as well. He doesn’t care how, he just wants him wrecked in all ways possible.
The other ship is Chilibal, I think that’s the most common name for it, and that one can’t even be called a relationship with a lot of fantasy.
Hannibal’s intentions for Frederick aren’t quite as malicious as Abel’s. Frederick has done him wrong by giving the credit for his work to somebody else, but well, it comes in handy for him, because that gives him a chance to continue, at least for another while. Also, he just doesn’t take Frederick seriously. He just wants to experiment with him a little, and maybe use him as a tool for this and that, just like he’s framing him for the murders in season two.
In situations where there’s intimacy between those two, it’s just like that. Frederick is basically a better sort of toy he uses to have some casual fun, or to “release pressure”, if we want to be a little spiteful here. Of course Hannibal is nice and polite as always, to make him believe there were feelings involved, but in all honesty, he doesn’t actually care much about Frederick. Breaking him isn’t his primary goal, but he wouldn’t care if that happened either.
TL;DR: Frederick Chilton isn’t allowed to have nice things. Ever.