Gacha games have mechanics like gachapon machines (those Japanese capsule ball machines where you insert tokens, turn the knob, and a little capsule falls out with a toy)-- basically they have like lootbox mechanics where you can spend some sort of currency (free in-game currency, or real life $$$) to get a lootbox that contains whatever featured item (usually a character) for the game.
I think if you add a mechanic where you spend some sort of currency to get a random hamster, then it would be classified as a gacha.
Nonetheless, thank you so much for playing! I am honored to have the hamster overlords grace me with their presence 🙇♂️