Revealing mistake: When Beauty is nursing from his mother, you can see that the foal used was a filly and not a colt when "his" tail swishes from side to side.
Plot hole: When Joe takes Black Beauty and Ginger away from Birwick Park to their new home, he is about 11 years old. When next Beauty sees him at the fair, he has aged to his early/mid twenties. This would indicate that Beauty had been at the home of the Lord and Lady Wexmire for about 10 years but not a single character in the household ages a day during his time there. His stay would appear much briefer than that.