Not necessarily. Someone with an H-1B or other "temporary" visa status allowing work would initially be issued a limited SSN card currently marked "VALID FOR WORK ONLY WITH DHS AUTHORIZATION". "INS" used to be where it now says "DHS". There are also one issued to people such as dependents who aren't authorized to work ("NOT VALID FOR EMPLOYMENT") but where it might be needed for tax purposes.
https://www.ssa.gov/ssnumber/cards.htm
Earlier I linked to the photo of one of the employment authorization cards. These are generally limited in duration and need to be renewed every few years. However, even someone with one of these wouldn't need to produce that SSN card. If you read the I-9, all they would need is some sort of photo ID (driver license, state ID, but not a foreign passport) and that employment authorization card.