Ha. You are not alone. The same happened to me. This is a very rare situation, and only happens to the 'fortunate' ones of us. I landed on Jan 16th and went to Service Canada to get my SIN on Jan 17. I was told that I could not get my SIN because there was an issue with my profile. I called CIC and concluded that it was an issue at the landing. I went back to the airport that week to see CBSA, and they found that, whoever did my landing did not click the 'Confirm' button in the system. The problem was taken care of instantly.
I suggest you do the same.
But there is a complication here. At Service Canada, upon finding that you were not in their system, did the person refer your case to the head office (by asking you to complete a paper application)? Assuming they did not, after you take care of the problem with CBSA, you can go back to Service Canada to get your SIN immediately.
But if they did make a referral and got you to complete a paper application, you will have to wait out the paper application process (4 weeks) whether you resolved the CBSA issue or not.
In my case, because the person made a referral (which I found out later, was irreversible), I had to wait 4 weeks for my SIN to arrive at my Canadian address (it would be sent to you by mail).