The Canadian license has a class rating, but it is separate from if it's a commercial or a private license.
For example, when I first got my Canadian commercial license, I hadn't yet done my multi rating so it had in the "class" section SEL - "single engine land."
After I did my multi rating (which is a stand-alone rating, I did mine in less than 3 hours) my commercial license then said, in the "class" section "SMEL" - single and multi-engine land.
However, if one does their multi rating with just a private license, then gets a commercial license (even if they did not do any extra training or flying on a twin) the class will still remain valid - SMEL.
The only time the class matters for the type of license is the ATPL - the qualifying flight test for that is an IFR ride in a multi-engine airplane.
Hope that helps a bit.
(I actually have a Canadian friend who made use of a loophole to get a TC ATPL faster by converting his TC CPL to an FAA CPL, then getting an FAA ATPL, then converting THAT to a TC ATPL but haven't heard of anyone doing the reverse.)