A common speech form in today's English is to make action verbs out of nouns.
In your example, wine, the alcoholic grape drink, is being used as an action verb. The phrase is saying "Drink wine when you want to regret your pony free childhood". Alternatively, it could be "Give me wine because I never had a pony". But not having a pony at any age is probably overwhelmingly common. I would go further that the phrase is meant to embody the culture of drinking wine regularly. This is probably understood by the people who have at least a glass every night, despite the quality of their days or their lives. They are saying that you should drink wine for the sake of drinking wine, but if you need an excuse, a pony-free childhood will do just as good as any. In light of the "pony for my childhood" meme, it's almost flippant. Most people don't really want a pony at any age. There's plenty of other phrases that can be found with a Google search of "wine. because", where the word wine is more of a verb than a noun.
An extremely popular example that you may better relate to is Netflix and chill. Netflix is a proper noun, but it's being used as an action verb. The sentence structure is laid out like an action verb. "Netflix" is not something you do. This speech form tendency is meant to capture what it means to watch Netflix, watching being the principle action of what you do with a Netflix account. This includes the on-demand television habit called "binge" watching that most if not all Netflix subscribers are familiar with. If you're still not convinced, consider that "Netflix and chill" is a common response to questions about plans for the evening. Someone asks you "What are you doing tonight?" And you reply "Netflix". Netflix is a proper noun, not a verb. You don't do Netflix. They mean to say "I'm going to just relax on the couch and let some Netflix show autoplay all night."
Consider some of these wine images, capturing the "wine drinking culture" meme:
Presumably, children whine, therefore this parent [drinks] wine. Wine is being used as a verb here.
"Wine" at the end is almost used as verb, saying "drink wine" or "give me wine".
An interesting twist on that old prayer. "I could use divine serenity and courage, but wine helps too."
Consider some of these pony images, accenting the meme "I never got a pony for my birthday":
"Mom and dad, gimme!"
Surely, you'd need a few glasses of wine first.
After seeing this, it should be clear that the saying isn't even about ponies. It's about wine. It's about drinking wine daily as part of your life ritual.
And finally, the less vocal "Drink beer" culture had their own take: