Is there anything that makes it illegal (or impossible) for an illegal alien (which could include asylum seekers whose applications are not yet approved or which may be denied) to have a Social Security Number?
Generally, an alien may obtain an SSN in three instances: (1) the alien is a permanent resident or has a legal status permitting employment in the United States; (2) the alien lacks such status but has a valid nonwork purpose that requires an SSN; or (3) the alien resides in the United States, has no evidence of alien status, and needs the SSN by law as a condition for receiving a federally-funded benefit. In this last case the alien may receive an SSN limited to a nonwork purpose. 20 C.F.R. § 422.104(a).
The SSA has said that it will not generally assign SSNs to aliens in the United States who are not authorized to work.