I started this game a long time ago and ended up playing for awhile and forgetting about it. I had a lot of pets though and now one of my dragons says "Heritage" with a red star next to it underneath his other info. What does this mean?

It means that the Dragon has Degenation 1 status. They now added this feature so it's easier for you to see which pets are Degen and which have GP (you will see 1, 2 or 3 green stars underneath the pet's name if its GP1/2/3).