If you want to have a colored name without any mods or plugins or command blocks just follow these steps:

1: Give your pet a nametag (It has to be a tameable pet!)
2: Make a team /scoreboard teams add [random name]
[u]3: Give your team a color /scoreboard teams option [team name] color [color][/u]
[u]4: Join the team! /scoreboard teams join [team name]
[/u]Now you have a pet with a colored name :)

[color=rgb(255, 0, 0)][b]COLORS
The colors are no color codes!!! doe it like this -> /scoreboard teams option [team name] blue (or any color you want)
black
dark_blue gold
dark_green gray
dark_aqua dark_gray
dark_red blue
dark_purple green
aqua red
light_purple yellow
white reset[/b][/color]