The book makes it clear. Chapter 5:
At the time he wrote these words, he was holding down a full-time job,
flipping Quarter Pounders at a McDonald’s on the main drag, commuting to work on a bicycle. Outwardly, he was living a surprisingly conventional existence, even going so far as to open a savings account at a local bank.
Curiously, when McCandless applied for the McDonald’s job, he presented himself as Chris McCandless, not as Alex, and gave his employers his real Social Security number. It was an uncharacteristic break from his cover that might easily have alerted his parents to his whereabouts—although the lapse proved to be of no consequence because the private investigator hired by Walt and Billie never caught the slip.
The Assistant Manager and Lori say he was a good worker, and they were basically short staffed.
Lori: "He was reliable, though, a body that showed up every day, so they didn’t dare fire him. They only paid $4.25 (four twenty-five) an hour, and with all the casinos right across the river starting people at $6.25 (six twenty-five), well, it was hard to keep bodies behind the counter.
Considering this is a pre-9/11 rural US with a short staffed fast food joint, I don't expect them to give much issue for a cook to have no ID as long as the information checks out. Not like they had E-Verify for SSA or IRS purposes back then. It's also not required to have or use the actual social security card present for a job (Form I-9 rules).
The script just glossed over it, as it doesn't mention the real social at all.