1. When you make your Tremorzilla pet in Following state, it would keep teleporting at you instead of walking at you first. For example, if you have a Tremorsaurus pet and you set it to Following state it would walks to you first and will teleport if you're going too far. But Tremorzilla straight up teleport at you even in a short range.
2. Tremorzilla pet will refuse to follow you if you're putting it in a sea/any liquid-based area, however it would actually follow you if you are holding Waste Drum item while they're in a sea, but they wouldn't follow you if you're not holding it despite have been put in Following state.