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.
Cave spinner webs are treated as solid terrain, which does mean that most movement effects can't go through them, just like they can't go through walls.
Levitation letting you go through the webs is actually not intended, thanks for letting me know about that inconsistency, I'll look into it.
EDIT: Ah, I see what's happening. The web effect is actually processing correctly, but levitating makes you immune to the rooted debuff. Perhaps at some point a 'webbed' debuff might be more appropriate here, but this is technically intended for now.
I don't think duelist with an assassin sword can blink through it nor can other blink abilities. I'd imagine the web from a giant mutant spider to be pretty resilient. you can burn it, disintegrate it, or throw something on it though.