There are three completely separate questions here, and as usual I have opinions on all of them of them.
Should you buy her a stick-shift car? Depends on whether she will be driving in congested traffic or moving traffic. Stick shifts are a real pain after an hour or two in LA-style stop and go traffic. If traffic is reasonable, she will probably like the stick. I have never known anyone who simply disliked sticks in general although they knew how to drive them. My daughters learned sticks at 16 and when they moved up to better cars, they both chose sticks. Others have expressed concerns about choosing subcompacts. I would echo those concerns.
If she learns in a new stick shift car, will she damage it? Almost zero chance of substantial damage. My daughters taught all their their friends in high school and college (at least a dozen kids that I know of) how to drive stick in my old Volvo wagon. All those learners combined shaved a total around 50k miles total off a clutch that would have lasted 150k miles otherwise.
Is it important that she learn to drive a stick? Absolutely! I was touring western Europe last month (Spain, France, and Italy) Since I have little interest in architecture, I found myself peering in the windows of parallel parked cars as we walked around the cities. Perhaps one in one hundred cars had an automatic transmission. The rare automatics I saw were in Mercedes and larger BMWs. Central/south America is similar. In Asia most of the private cars are sticks but the tourist rental cars are automatics.
When my kids were born, I made a mental list of all the skills I would make sure they would learn before they left home, just in case someday they really needed to know. Among these were: Swim, drive a stick shift car, perform routine car maintenance, ride a motorcycle, groom-saddle-ride a horse (though we live in the city), assemble and fire several types of firearms (though we own none).