Sweet! Time to talk about the complicated history of sexuality, slavery, morality, and all that awesome stuff. From a historical standpoint! Woo!
Alright, so the first thing that I see wrong with your post is the opening argument itself. "Slaves cannot give consent." Well, no. That's actually provably not true. Slaves, despite their living conditions are still people with desires, needs, emotions, and all that good stuff. Honestly, that opening argument itself just confuses me since I think we can all agree that the reason why slavery is bad is because it takes away the personal freedoms of an individual in various ways, and yet here you go taking away even more. Of course a slave can give consent, if they want to.
There are actually many examples in history of true romance between master and slave. The most dramatic (in my mind anyway, I'm sure people will of course disagree) was Suleiman the Magnificent and his slave turned empress Roxelana. By every account we have of them they seem to be nauseatingly in love. Dude wrote crappy love poems to her, had this whole succession crisis because he wanted one of her sons to succeed him. It was a big thing.
Anyway, so, first point, no. Doing the nasty with a slave does not automatically imply rape. Saying that it does with 100% certainty just takes away what little freedom a slave actually had.
Not to say that slaves weren't raped. They totally were. Happened all the time.
Next point. Reading the passage itself "Ill-advised liaisons" well, ill-advised by who? Certainly not Azir, since he's getting his groove on. There is really only two options: by Xerath or by the Shuriman people. Now personally, I'd vote for the Shuriman people, but it's ambiguous. Now, what would make a liaison ill-advised? If he's just raping slaves, well, people would have been doing that for years anyway, that doesn't make sense to me. It could be because he was actually something of a don juan to them. A great woo-er of women, which personally I think makes much more sense to his character. He sees slaves as people, like Xerath his closest friend. But he is still a charismatic intelligent emperor, and throughout history those kind of people seem to get others willing to bone down. But that's really up to your interpretation, if you want to imagine him being a rapist go for it. It's certainly possible.
And final point. If Azir is totally a rapist, if he used and abused his slaves. Then I will finally congratulate Riot on creating a truly realistic morally complex society and the effect it can have on an individual. Really, all their other efforts to do that have been pretty much crap. Here we can see an actual realistic representation of an ancient emperor and what that actually means. A real historical figure and how good and bad they can be. How the culture changed them into something so warped and foreign to our modern sensibilities. That's what a historical figure looks like. That's what our past was. No great empire existed without slaves before the industrial revolution. The cultures that had slaves and harems tended to have a lot of cross-breeding that was not pretty to our modern sensibilities. That is history and I applaud Riot for having the guts to do that.