This is the dreaded stabling publish. It went live on Sonoma and Siege Perilous a week ago. The number of pets you can have stabled depends on your total displayed skill spread between taming, lore, and veterinary. The maximum you can have stabled is 5 pets at any time, regardless of which stablemaster you use, and only if you have 240 points used between those 3 skills. You can claim your pets now from any stablehand, and claim them specifically by name.
All invulnerable NPC's have yellow names. This includes stablemasters, player vendors, and some regular vendors.