I've recently tamed two cats, and they teleport to me when they're really far away, but they do not follow me when I'm at a medium distance - instead they just sit frozen, and keep teleporting to me when I get very far away. When I get close to them, they look at me and rotate towards me (without moving position), but when I move further away they just sit frozen and do nothing.
[Edit]: Upon further testing - if there's a bed or a work station nearby, the cats will climb up on it, but will still not want to follow me.
It's the first time I've tamed any cats in this world - and I'm currently playing vanilla 1.15.1
This world was created in verison 1.11 - (created on the 6th of December 2016), and this village was also generated in version 1.11. The world has been through all the versions - 1.12, 1.13, 1.14, and currently in 1.15.1. Not sure whether this makes any difference.
I've never had any mods installed - only had one texture pack which I no longer use.