Yes, but the Canadian Trakehner Association is recognized by Agriculture Canada and is the official association for the registration of Trakehners in Canada. Their administration is overseen by the Government of Canada. The studbook may be smaller and lesser know, but by many standards certainly not a lesser studbook.
A Canadian Trakehner is a Trakehner registered with the Canadian Association…other options are ATA or the German Trak Verband…and there are probably more. I’m not a Trak expert by any means so don’t know how many international associations there are.
Some people are very particular in which studbook the horse in registered. It really is a personal preferences. Some are somewhat more particular when choosing for broodmares and stallions…but the good ones can be transferred to other books without too much trouble. It you use your horse as a riding horse, not a breeding animal, the registration certificate serves of proof of pedigree…and the specific association that issued that paper is for most not all that important.