This is what is known as an erkennungsmarke. Roughly translated, it's an ID disk, similar to the dog tags issued to American soldiers. These were issued generally by the replacement battalion with which the soldier trained; unlike the US military, every division maintained its own training unit which fed directly into the parent unit. In this case, SS-Panzergrenadier-Ausbildungs und Ersatz-Bataillon 10, attached to 10. SS-Panzer-Division "Frundsberg." Presumably, he was sent on to the division after completing his training. In addition to the name of the unit, it includes his blood type (A), and a unit ID number. Unlike American dog tags, the soldier did not receive a general service number unique to him; he received a number within his unit, usually three or four digits.

Before you start suspecting your grandfather was a secret Nazi, bear in mind that 10. SS-Panzer-Division was formed late in the war, largely from teenage conscripts from the RAD (Reichsarbeitsdienst). However, 10. SS-Panzer-Division did not fight in the Ardennes; it was engaged in Normandy and at Arnhem, then withdrawn for rebuilding and committed to the Eastern Front in the last months of the war. Either the family lore is wrong, or he transferred out of the unit some time after completing his basic training.