I think the issue that has been raised is the question of "one of the players has severe arachnophobia and DM planned to run an underdark/Drow/Lolth based campaign". The player isn't asking for an unreasonable request, but fulfilling that would ruin the DMs planned game. Whose preference should win out?
To hear all of y'all talking, the DM gets his way and the player gets to stepping.
That one is easy as it requires minimal lore tweak on behalf of DM to accommodate player. Instead of having spider fetish, Lolth now has bat fetish, drow are all about bats, it fits thematically with Underdark and everyone is happy.
If DM knows in advance that one of the players has arachnophobia, they would probably skip drow themed campaign in the first place or would do above mentioned lore changes.
If they don't know, player should communicate it with DM before campaign starts, and talk it out with DM. FE:
DM: Hey peps, you interested in drow centered Underdark campaign?
P: Sorry, but i have severe arachnophobia. Can we do it without spiders?
DM - option 1: Sure, i'll make some changes so no spiders are present.
DM- option 2: Sure, no problem, we can do something else.
DM- option 3: Sorry, don't have anything else and spiders are kind of Lolth/Drow thing. Any suggestions how we can work this out?
DM option 4: Sorry, no can do, don't have anything else. Maybe you should skip this one.