Hi Annie
It was all prefectly normal. Quebec was British, as was Ireland - so he joined the army - there was only one.
There is a good possibility that his father was a soldier serving in what became Canada, and then moved to Kinsale - but there are obviously other possibilities. You could try the regimental BMDs on sites such as Findmypast. They are not complete but they did start before civil registrations. Later in the century the army would allow boys to follow their father until they were 14, so it may have been earlier in 1812. The 6th were in the Peninsular around that time so maybe his family were not selected to go, and it was either be enlisted or be sent to the Workhouse.
There was at least one other James White in the 6th Regiment (from 1808 to 1809 at least) as he received a medal with four clasps for battles fought in that period.
Ken