In THIS part of character creation, you are picking your biological race and subrace. Who you worship, and whether you are a good or evil aligned individual should NEVER be baked into this particular choice.
I think they should have used the term Underdark rather than Lolth Sworn. It makes sense for this to be a separate tag because the cultures of Underdark and the surface are completely different and would definitely have some influence on how you would react to things, changing the name would remove the implied alignments and deity.
Yea I agree that's not what I meant but I didn't write it well. To summarize if you're a drow you're a drow and everyone should see you as a drow. There's not a good drow subrace and an evil one.
People should definitely treat you with caution no matter which Drow subrace you are. Only the player and perhaps perceptive Drow would be able to determine where you came from. The subraces should be just for player dialogue options.