I have heard mentioned of this and just wondering how this is achieved? Thank you.

The pet ID number is tied to the pet itself, but what you can do is swap names between two pets if you own both of them.

https://subeta.net/pets_name_exchange.php

What Laurey said; if you have two pets that you want to trade all the information on. It costs 750 csc to do that.

If you have one pet that you just want to change the name on, you're better off using the drop-down menu next to their name on your Pets page and changing the name of that single pet, which costs 500 csc.

Ah, I own both pets but don't think the 750csc option would also change the pet's colour, etc cause I want to transfer my angelic feli to the lower id number and keep all her info as well.

You're right, it wouldn't. The only things that would change are their names, so you'd have your Angelic Feli with the same profile/stats/ID but it would have a different name if you swapped them.

Ah, will have to think on it. Not really wanting to save up for another potion since I don't plan to keep the other pet, will have to price everything and decide.