I think the confusion comes from the fact we've seen moments where Nikolai visibly hasn't had his cloak on him and him still using his ability, for example when he's disguised. Those moments could be explained by the fact he probably had the cloak hidden under his clothes. Though that then doesn't explain how he did that trick to fake his death during the sunday tragedy where he didn't have his cloak on him. I think how his ability works is that he doesn't necessarily have to be wearing it, just that his cloak is a /literal/ portal (basically his cloak can act as either the exit or entrance for the connection he wants and is also the setpoint for that 30 meter distance he's restricted too, meaning that his ability is to activate his cloak's spatial connection to a pocket dimension so that he can use his cloak to set spatial connections within 30 meters of where the cloak is without having it on him. This is then why he was able to do that trick in the sunday tragedy, get fyozai out of their cells and to hold himself up with that bar by using his ability by taking off his cloak and placing it on the ground (the cloak opened a connection between the area he placed it down in and the area where he's currently holding himself up).

It is exactly as how he said it in that one chapter when we're first explained his ability, he does need his cloak because it's what's connected to that pocket dimension, but he doesn't necessarily have to be wearing it since he never specifies that.