Use bleed to make it easier to tell his location

If you're struggling to track him or dps here a piece of advice : Use the bandit's set combined with the devastator and the spitfire. His weak point is big enough to focus it with the crossbow and up your mods with the nice damage input. Then use the bleed and the fire to deal an impressive amount of damage. It also makes it easier to track as said in comments. But you can not inflict bleed or fire status while he's invisible (it never work for me at least). To speak about the two different types of illusions, the blue ones are inoffensive until they reach you in close range. Try to kill it with a single bullet from the spitfire and just walk perpendicular to the blue projectile it launches. It's enough to dodge it. The violet illusions manage to combo in close range and they are doing more hits according to the boss health (the more low it is, the more combo hits they will do). Two dodges and a sprint in the opposite direction are enough to avoid taking damage. While the boss is in the aire, if you got a bleed spike and your flamethrower max stack, you won't need help to bring him down alone even in a three party member.

You can also mark (MMB) him as he's going up to the air, and the mark sticks to him after teleporting.

Summons autotrack the real Dream Eater, use Turrets and Swarm.

Worst boss ever designed. This boss glitches so much, aint fun anymore.

If you struggle to tell which ones him when he flies into the air, pop hunters mark at the centre of the room, he's the only one not outlined

I noticed that when he flies into the air, when you pop his illusions the real one flashes for a second. And you can see the flash through walls, which helps.

Also, Spitfire is a godsend on this fight. Not only does fire mark him when he flies into the air, but his illusions have no HP so a tiny burst with it will drop them. Instead of needing to run and dodge his illusions, you can farm them for ammo while chasing him.

Hit him with the devastator alt or any other Damage over time alt fire.
That way you´ll be able to easily keep tabs on him once he goes airborne

Using hunter's mark will mark all copies/illusions except for the actual boss