@HighRollerHydra  
In the Letters pages of Marvel Comics, there’s this thing called a No-Prize, awarded to anyone whose letter explains away an apparent continuity error with logic, in the spirit of the comic.
 
If this dog is Winona, there’s a continuity error (age) and an animation error (three legs). I just eliminated both with one easy and probable fix, while using the show’s penchant for real-world names to name the new character and “correct” the name of the old.
 
It’s my attempt at a pony No-Prize explanation, and it’s a lot more fun than, “Oh boy, the animators screwed up again. Good thing it’s just a show for little girls.”

@TheMasterEraser  
Or it’s a cartoon.

@HighRollerHydra
 
@Konata Izumi Pony
 
I disagree. I don’t believe this is Winona (or Wynonna). Here’s why:
 
Fact 1: The dog in this photo was young when AJ was just a foal.  
Fact 2: On our Earth, dogs are considered old at 10, and usually die by 13.
 
Applejack is a young adult, approximately the same age as Twilight, who has been in school as long as Spike has been alive. The usual fan estimates of their ages are 15-25, with Spike at 5-15. If AJ is at least fifteen, it’s unlikely the pictured dog would be nearly as spry as Winona, who leaps over cows and is generally quite active.
 
My headcanon is that this is Winona’s mother, a three-legged dog named Naomi, as in Naomi and Wynonna Judd, the famous American country music mother-daughter duo, and most likely the inspiration for AJ’s canon dog’s name.

@Konata Izumi Pony  
Most likely.

Adult winona have four legs, when he was younger he had 3? Animation error?