It isn't any single mechanic, it's how long the fight is (much longer than the Omega fights) and how low the margin of error is for each single attack, and how deadly each single attack is.
Several of them kill you instantly either directly or by proxy -- they either knock you off the platform (instant death) or are like the fist pound (damage high enough to insta-kill a healer or DPS barely geared for the instance). The Tail slaps are almost as bad, and again, insta-kill if the platform was already cracked and they didn't notice the subtle indicator (which can be hard to see in such an intense fight).
Sometimes his dives across the platforms are painted in a way that they are physically impossible to move out of before he does it depending on where you were standing before he chose to do this, so you either eat it or get pushed off (again, instant death).
With all the spells and effects going off, there's an instant or near-instant death possibility during the adds phase -- Meteors+Death Sentence is pretty much a guaranteed kill on the tank and/or anybody else caught in it and sometimes it is difficult to see the marker for that too with all the other markers on the ground for the meteors.
There's the Earth Attack where Shinryu paints 2 people... if those 2 people are derpy, they get the entire raid caught in the blast with no chance of escape (making it so that 1-2 players can cause a wipe by not doing something right is kinda ridiculous in a fight that's mostly meant to be Q'd for with randoms) depending on where they happened to be standing when the thing paints the AoE area.
There's just so many attacks that will either insta-kill you, or put enough hurt on you that healers just can't keep up with it sometimes. That's what makes Shinryu more difficult than Omega bosses -- it's just AoE after AoE after AoE after AoE with hardly any time in-between to recover if a few people make a mistake. Shinryu requires all 8 people to be on their game pretty much, or very close to it. You could maybe get away with a derpy DPS who lays on the ground dead most of the fight, but if you get a 2nd... chances are, unless the remaining two are entirely awesome and over-geared, you're not gonna make it.
And God Forbid one of the derpy people be a tank or a healer.... then you're just done, period (assuming you're not at like 5% health or something remaining). A 320+ healer could maybe handle that, maybe. But certainly not one who is 280-300.
And also, those cheap auto-attacks on 2 random people in the raid no matter where they are I think are the icing on the cake. Wholly overkill. Even if you do barely survive some AoE... chances are, you're dead before a healer can get you healed because of the stupid Auto-Attacks which tend to land <1sec after major AoE attacks.