You can assign the breeder brand or Iron (AKA burn mark) to each horse and it automatically appears on pedigrees.
To do so, you need to first add the brands/irons to the list of brands in Admin/Configuration/Branding or Iron (at the bottom, the last option on the left).
Then, you edit the horse and set the brand/iron of its breeder.
If you access the horse profile, you will see it as one of the info pieces:
And in the pedigree:
It appears also when generating horse PDFs: