I think this is a great idea. I’m not very familiar with Bedrock, so I’m not in the place to criticize. But I think I can try my best to reply at the very least. :)
Most servers with cosmetics such as pets or companions have a naming system, so I think it would work perfectly. I’m also confused whether you mean recolours of pets or coloured names, but if it’s about coloured names then yes. I think you should be able to use formatting along with naming the pets too. I don’t think recolours would work well, but who knows. Of course the names would still have to be appropriate, as you mentioned yourself.