The customer is claiming their "verification process" is not possible to satisfy, and customer service is non-responsive. If the customer's story is true, the only issue is stupidity on the part of Freelancer.com. All the CEO said is the customer "failed to complete our Know Your Customer (KYC) process" -- ignoring the customer's claims that the process is completely broken and customer service is not providing a way to meet the requirements. If there were more to it, I think Freelancer.com could get away with something like "The customer has not submitted what he said he did." In the absence of that, I think it's reasonable to assume the customer's story is correct and honest, and Freelancer.com has really bungled this and is only digging their hole deeper with form-letter responses on social media.
As for privacy and legal liability, he's already confirmed that someone is a customer, revealed the location (New York) of an employer of said customer, and divulged details about their response to the employer, so ducking behind "privacy" when the crux of the issue is questioned, rings hollow to me.