Which hero are you? (Game)

Not alch, but I can understand the confusion after reading it over again, pretend that the word mount is pet.

Abaddon?

Not abaddon, however the realm of "Zeus's friend" does have to do with Avernus (has nothing to do with the hero in the riddle)

Chaos Knight

Not chaos. (I think I made this too hard :/)

I didn't really take the time to decipher the full riddle and looked at it somewhat literal, I scanned it and what I got from it was that the said hero had traits or was proficient in:

• Mobility: "I'm just too quick!"
• Longevity or Everlasting: "I'll make you eternally sick." (He is a force of the universe, therefore unless KotL is abolished he is eternal.)
• Mount in DoTA and Dota2: "My old mount was constantly tripping"
• Is proficient in killing (Death): "Now Zeus's brother is my friend, Because of his River Lethe." (Hades)
• Confusion: "Alright now you know who I am,
  I will try to make you forget.
  Now guess who I am" (When you find him he will be real, he'll activate Phantasm to disrupt targeting.
• Dueling Potential and / or high Damage Output: "Quick now! Before your fate will have been met."

My other guesses were Weaver, Chen and Disruptor, I removed Weaver from thought because he has no steed. Chen was removed because it was a rough fit overall into the riddle other than a mount, the ability to remove someone from the fight to confuse, and the word fate. Disruptor also had a mount, and glimpse for all these clues on forgetting, as well as the mention of Zeus because of his affinity for electricity.

From reevaluating the riddle I think its Phantom Lancer. He's difficult to find due to his illusions, Invisibility, and movement speed (bonus from drums/yasha/manta). I think the mount was pissed because there are so many Lancers to carry (ironic, lol). Zeus' brother can refer to Poseidon (Azreal/Azwraith is a fisherman) or Hades as water or death because of the referenced river that hints towards more confusion / forgetfulness. And the last 4 lines sounds like the usual gank on Phantom Lancer.

Well, I'm impressed that you deciphered it so well, but still wrong. :P The biggest hint I gave was the River Lethe, which you mentioned was the river of forgetfulness/oblivion. Any last guesses before I tell the answer?

I'll take that as a no there aren't any guesses left. The answer was Pugna!

You'll never find me,
I'm just too quick!
These two lines refer to pugna's ludricrously high movement speed

But if you do,
I'll make you eternally sick.
Pugna has to do with death... And stuff... And things...

My old mount was constantly tripping,
'Til it left when is got pissed.
This is where is probably got tricky. As I said, if you replace the word mount with pet, it references viper, who is always tripping (on acid)

Now Zeus's brother is my friend,
Because of his River Lethe.
In the underworld, the River Lethe is the river of oblivion and forgetfulness

Alright now you know who I am,
I will try to make you forget.
Again, the theme of forgetfulness and oblivion

Now guess who I am,
Quick now! Before your fate will have been met.
Referring to Pugna's high burst damage with decrepify.

Sooooo... I suppose that xCO2 can go, he did give similar reasonings.

Well I guess you go again or something? Since noone really got it? I don't know.

Well, I can't think of any right now. If someone else has one say it, if not I'll be back later when I think of one.