1.21 Neoforge & Fabric port of the original Wandering Pets mod.
Wandering Pets (Updated) is a fork of the original Minecraft 1.16 Forge mod, reimplemented for Minecraft 1.21+ NeoForge and Fabric, while also including a few tweaks and fixes.
This is a very simple mod that allows players to toggle the behavior of tamable mobs between following and wandering (as they'd do when not tamed), so they don't just sit down when not following the player. To use, simply Shift+Right-click your tamed mob while making sure they're standing, and they'll start wandering around.
As for cats, because they also sit on objects and sleep, there was adjustments to their behavior so they'll alternate between roaming and sometimes sitting or sleeping. You can turn this off in the configs, but that'll make the cats really sleepy.
wanderingpets-common.toml in config tweak mod settings