Why can Phantom Assassin dodge Omnislash with Blur's active?
Solution 1:
The active effect of Phantom Assassin's Blur - that is, the invisibility even while attacking, using items, and under true sight - is dispelled 0.75 seconds after moving within 600 units of an enemy hero, not instantly. Since Phantom Assassin is invisible, Juggernaut's Omnislash can't find a valid target to continue attacking and ends.
Keep in mind that after the rework of Omnislash, the damage is no longer physical spell damage, but Juggernaut's autoattacks themselves. This means they can miss, and Monkey King Bar does not grant true strike anymore.