There are lots of options for labelling by the horse if you want to go this route. Something as simple as different colored piece of tape (electrical tape or duct tape) on the same buckle on all blankets, color designates size or horse. You can get tags to hang on the buckle, etc.
Since all the blankets and horses are yours I get why you want to sort by weight. When mid weight weather comes along you open this one bin, pull out all the mid weights and figure out which blanket goes with which horse instead of opening a bin for each horse and figuring out which blanket is the mid weight in each bin.
In a boarding situation it makes more sense to sort by horse.
Edit to add - this is just the first example of what came up when I Googled Horse Blanket Labels.
https://www.etsy.com/listing/1112283714/equine-blanket-tags-single-sided-custom?gpla=1&gao=1&&utm_source=google&utm_medium=cpc&utm_campaign=shopping_us_e-pet_supplies-riding_and_farm_animals&utm_custom1=k_EAIaIQobChMI-4id9t61_QIVrsqUCR13gATSEAQYASABEgI1JPD_BwE_k&utm_content=go_12559942930_120251256420_506995551488_pla-353481983254_c__1112283714_12768591&utm_custom2=12559942930&gclid=EAIaIQobChMI-4id9t61_QIVrsqUCR13gATSEAQYASABEgI1JPD_BwE