I do shoot Canon and have used their software. As far as I’m aware they effectively use an icc files on all of their cameras that will shoot raw.
The Adobe profiles do differ but I’ve not found that to be a problem. They did tend to bring in more highlight than Canon did but that was some time ago and may have changed due to the number of stops wars in jpg’s.
Photivo comes with a lot of camera profiles. That can be a good source at times if it has one specifically for the camera that has been used.
As you mention though the solution is to use the same profile in both applications.
The other problem can be that the manufacturer’s software may play with colouration. That in some ways put me right off shooting Nikon.
John