pony

It is a small breed of horse. It is also a small glass adequate to hold a single swallow of whiskey. I would like to have the latter.

It is a small horse rather than any specific breed. It just means any horse under a certain size (14.2 hands or 58 inches high).

Is it something that kids want as a present in America?

I'm watching a American TV play and there is a lot of mention of pony when talking about desirable things

Lots of kids (especially girls) are 'pony mad' and want one, yes.

To expand on that a little, it's also a highly impractical thing to have or to want.

So sometimes "a pony" is used as an archtype of what you want but can't have.

It's sometimes used jokingly because it's SO common for little girls to express a desire for one ... for example, the single uncle will joke to his brother that he's going to get his niece a pony - and implication is

GrammarGeek,
I like your pragmatic way you see ponies. That's exactly as it will be, in most cases anyway.
Kajjo

thank you, no wonder I hear the characters in FRIENDS joking around about have a pony