Me thinks there is missing info here.
1) I guarantee that taking a drug test is mandatory for employment...and you signed it.
2) They do not have to have a reason to do the test, and it is highly-highly likely the papers you signed also said so. Refer to #1.
Here is the policy directly from Starbucks company. There is nothing in the policy regarding drug testing.
Substance Abuse and Weapons
Starbucks has strict standards regarding substance abuse and weapons. Partners are not permitted to use or possess alcoholic beverages on company property, except where alcohol is specifically permitted at a Starbucks-sponsored social event. An exception applies if your job involves the sale of alcohol, but in that case, you may not consume the alcohol and must participate in any special training required for that business.
You also may not use or possess illegal drugs or controlled substances on Starbucks property or while you are engaged in any job-related activity. Partners may not report to work under the influence of alcohol, illegal drugs or controlled substances.
Partners may not have or possess any weapon while in a Starbucks store, plant or on other Starbucks property. Starbucks takes its rules regarding workplace health, safety and security very seriously. It is essential that you understand and follow them, together with any more detailed guidance provided to you.
http://globalassets.starbucks.com/assets/eecd184d6d2141d58966319744393d1f.pdf