I really want to help you out here so I will do the best I can, especially given my limited knowledge of ponies vs. horses.

Records indicate that during the second half of the 19th century something like 3% of Jews worked in the field of transportation: coachmen, carters, porters, etc. Jews were pretty dominant in the field; by the end of the 19th century, about 21% of independent coachmen and carters were Jewish. This means that quite a lot of Jewish people in Poland (among other areas in Eastern Europe) would have owned a horse for work. This figure doesn't include others who, though they did not work in transportation, would have had a horse for work (for example, Tevye from Sholem Aleichem's "Tevye the Dairyman" has a horse that features prominently in the stories. He's a milkman, and he travels with a horse-drawn cart to deliver the milk). The nature of Jewish involvement in transportation would change during the 20th century, as transportation became increasingly mechanized. Nonetheless, horses would continue to be used for work and transportation, especially in smaller towns and villages (if you look up "horses" on the YIVO website, you'll find a bunch of pictures from market days and other moments capturing regular prewar life with dates up to the 1930s).

You'll note that I've only mentioned horses, not ponies, and horses that were working horses and not, presumably, someone's pet.

I'm going to make a series of educated assumptions about Manya in order to determine how unusual it would have been for her, her sister, and her cousin (and others) to all have ponies. First, I assume that the character is approximately 81, the same age as the actress Rozsika Halmos was when she played her. The episode aired in 1991, which means she was born around 1910. Because there is no mention of Manya being a Holocaust survivor, and because she has an accent, I'm also going to assume she came to the US at the last moment, say 1938 when she would have been 28 years old. This means that she would have missed the presumed peak of Jewish horse ownership in the late 19th century. This does not mean that her family would not have owned a horse, but it does make it more likely that owning not just one, but at least two horses would have been more unnecessary, and therefore more unusual for the time period.

You'll notice again that I keep referring to horses, not ponies. I didn't find photos of ponies the way I found some of horses in the YIVO archives. And based on my cursory search to figure out what you would use a pony for in general, I have to conclude that it wouldn't have made much sense to have a pony helping with your work when you had horses available. None of this means that there were no ponies in prewar Poland, just that it does not seem to have been at all common for people to have had them.

Conclusion: Manya's family and village would have been very unusual in having so many ponies. Owning a horse would have been more common (though less common in the 20th century than it would have been in the 19th), but even if we're talking about horses it would still have been highly unusual for it to be Manya's horse or her sister's horse, or her cousin's horse as opposed to a horse owned and used by her father for his work.

Edit: After doing some more research about horses in Poland!, I take back my statement about how rare ponies must have been, and how it was unlikely they were used for work. My original answer as stated in the conclusion still stands, but it is more possible than I originally thought that Manya's family would have had a pony.

Sources: Mostly https://yivoencyclopedia.org/article.aspx/Transportation. I recommend also searching "horse" on the site; you'll get some neat pictures.

There's also lots of Yiddish literature that features men (it's almost always men) and their relationship with nature/their animals. I mentioned Sholem Aleichem's Tevye the Dairyman, but I would also recommend checking out Mendele Moykher Sforim's work.