Lloth specifically blesses her followers with red eyes to mark them as 'hers'.
No, she doesn't. That's the problem.
Bingo.
Here we have
Vhaidra and
Nathyrra-two drow assassins. One of them follows Lolth, the other Eilistraee. Not the sort of distinction you'd want to mix up, and not one you can tell by the color of their eyes.
For a more extreme example, mistaking
this violet-eyed drow woman for a 'seldarine drow' would be just about the worst mistake you could possibly make. That's Eclavdra, Lolth's highest-ranked mortal follower in D&D. With the same eye color that 'Seldarine' drow get by default in BGIII, same eye color as the famous drow rebel Drizzt.
But besides that, ignoring all the non-red-eyed followers of Lolth, we should think about what the ramifications would be if every drow who turned from Lolth had his or her eye color change. They'd be killed on sight if it happened in a drow city. Because in a society like Lolth has built where religious dissent is punishable by death or driderhood, an obvious physical distinguishing characteristic like that would effectively be a death sentence. Similar to one of the criticisms of the LP series in which a bunch of drow worldwide spontaneously had their skin color change to mark them as no longer 'tainted' by Lolth. An express ticket to your own funeral.
