How can one verify if the FCC/CE certification mark on an electronic component or product is genuine?

Search the FCC database by their stated FCC ID code: http://transition.fcc.gov/oet/ea/fccid/ The first 3 or 5 characters is the Grantee Code (assigned to a company).

There isn't an equivalent for CE.

CE can be self-certified, so there's no central register of products.

The manufacturer is still obliged to keep records of the certification process: they should be able to give you a copy of their report.