This is working as intended. Your pet has 6 Tiers of friendship with you. As you become friendlier with your pet they will eat less. During that initial phase of taming a new pet the first hour is a bit annoying but if you feed it consistently during that period the Pet will tier up and require substantially less food. Once they're Tier 5/6 you barely need to feed them at all. Tier 1/2 are the rough Tiers.
Think of it this way, you just tricked a wild animal and its not too pleased about that. Feeding it yummy treats makes it trust you more. But if you stop feeding it right away, or heaven forbid let it get killed, that goodwill goes down the drain.
But yeah you want to be prepared when you tame a new pet with at least a few stacks of food, you can figure out what food the pet wants and lots of other Hunter Pet information here;
https://www.wow-petopia.com/classic/