Is that based directly on previous choices, or is it a question of whether you have enough Paragon or Renegade points to unlock the "I WIN" button?
Either way, I would prefer if any Shepard could attempt to persuade him, but some Shepards would simply fail and Mordin would release the cure anyway.
But that's a different issue.
Well, in ME3 it's Reputation points that are critical. Not a great mechanism since it's a completionist check, but way better than ME2's abomination. I believe the requirements for this check are so low that any Shepard can pass them. Similar to talking down the VS on the Citadel, where almost any Shepard who's allowed to make the check will pass it; only a non-import Shepard who does almost no optional content can fail. Main plot Reputation checks seem to have been set low.
But success is dependent on previous choices, yes. Specifically, Mordin will only back down from releasing the cure if he sees current krogan leadership as hopeless, so both Wrex and Eve have to be dead. If he won't back down, any Shepard can shoot him.
Well, except that revealing the sabotage plan earlier in the game will lock out this entire interaction; it's presumed that Shepard isn't trying to sabotage the cure. But with the sabotage plan revealed to the krogan leadership, actually going through with the sabotage would be crazy anyway.