My son has got into similar problem. Applied for SIN in Feb 2024 and it's Jan 2025 now - no SIN yet, we were calling multiple times 1-866-274-6627, where they just say it is in progress. Can't understand what's wrong and if anyone is actually working on the application, no indication of a progress. We even got officer from Service Canada talking to representative answering 1-866-274-6627 and this vicious circle is not broken still.
My son is Canadian citizen, finished school here in Canada. Without SIN number he can't start to work and the institute he has applied to requires the SIN number. He can't run own business either, so basically he has got stuck in limbo. I'm thinking about a court case against them.