It must be considered solid terrain and not a hazard. I assume that this is treated like a physical feat, rather than a supernatural one, so the monk would get stuck in the web trying the dash through. The animation, however, uses the same as levitation, and you can levitate over the webs when levitating from a potion.
I interpret use of this animation of the monk "flying" as acrobatic flips and maneuvering to avoid traps or stuff on the floor. So is the web stretching from floor to ceiling? If so, why can you levitate through it? Seems inconsistent.
It makes sense that if you selected a tile past a web that the game would tell you that you can't make it there instead of just throwing you into the web without warning.