Most likely Service Canada agent that you encountered was a newbie.
- Confirmation of Permanent Residence (COPR) issued by IRCC
Note: If the COPR is used within one year of becoming a permanent resident, it is acceptable. After this period, the Permanent Resident Card is required
print this webpage and bring with you:
https://www.canada.ca/en/employment-social-development/services/sin/apply.html
> what did you say to the agent that made them allow you to get a permanent SIN with an incorrect name
didn't explain anything. told them that govt messed up my name, asked if SIN itself could be issued in accordance with passport.
They responded by saying they only look at IRCC records. It was kind of a common thing for them as I was not the first person in that month that had this issue.
So what they did was that they applied legal name change procedure to change my name from the correct one to the incorrect one, and asked me to come back after it was fixed on ircc side.
I think DOB is even a more minor situation than the first name. Look at this person:
https://www.reddit.com/r/ImmigrationCanada/comments/qkrb2z
He also got his SIN. So you should be able to get it as well. Even with incorrect DOB, it doesn't affect immediate things. (just don't renew your health card yet).
Try multiple Service Canada location. Even better call service Canada before going there, asking agent to confirm if they can issue a SIN that way, with status verification doc.