Generally, knowingly and willfully giving false information to certain entities is fraud and is illegal. Especially when the form has you sign off that the information provided is correct to your knowledge under plenty of perjury.

Even if you gave a fake SSN to, say, a landlord (they need it to run a background and credit check on you and to better chase you down if you don't pay your rent) that would almost certainly be considered either fraud or theft by deception.

Generally, you provide the entity your SSN because they have a legitimate need to identity you, and that includes identifying you as you outside their internal system.