If you are worried about the service Itself then there is nothing to be worried about as Its the first fully licensed bitcoin exchange out there[1] so It's not a scam.
Now , for using your ID and If they are not going to use anywhere else ? I doubt it. I personally don't believe that any big company out there asks for personal information's without using them with one way or another. Still , It's up to you to share your ID or not. It kinda defeats the whole purpose of using bitcoin though , I mean bitcoin is all about being anonymous and keeping your private life.
[1]
https://www.bitstamp.net/payment-institution-license/Actually, your link from the footnote leads to a 404 page of Bitstamp.
404 error
...aaaaaand it's gone.
Page not found. The requested url was not found on our servers.
Bitstamp claimed to be licensed in Luxemburg for years but has recently removed
all references from their website. In another thread a user pointed out that only
one of the Bitstamp corporate entities has the payment institution license, but
it is not the one where most users are registered. It is proving to be very difficult
to obtain accurate information about this.
However, Bitstamp has been the best exchange in my opinion so far and I have been
using them since 2013.
If anyone is looking for another alternative with a valid license, bitFlyer has recently launched
in the US and Europe as well after originally serving only the Japanese market.
They have also been granted a valid payment institution license from Luxemburg recently:
https://twitter.com/bitFlyerEU/status/955726910427820034