Not exactly. You have one pilot certificate, and in this case it would be a commercial certificate. You can't be both a commercial and a private pilot.
Under section II of your license it will say "COMMERCIAL PILOT".
Section XII RATINGS will list: "AIRPLANE, SINGLE AND MULTI-ENGINE LAND, INSTRUMENT AIRPLANE."
However, under section XIII LIMITATIONS you'll get something to the effect of "SINGLE-ENGINE LIMITED TO PRIVATE PILOT PRIVILEGES."
To remove the limitation, you will need a single-engine checkride, but your student is technically correct: you'll be a commercial pilot AS&MEL.
I had a single and multi private ticket. When I upgraded my single rating to commercial I got all excited when I watched the examiner fill out "COMMERCIAL PILOT: AIRPLANE, SINGLE AND MULTI-ENGINE LAND, INSTRUMENT AIRPLANE" on my temporary certificate. It was short-lived, though, and he dashed my joy when he filled in the limitations section.