First and foremost, of course alcohol influenced european culture through the centuries and this is noticeable at different levels, including the language. Essentially, alcohol combines two of the most complex symbolical elements: water and fire. The beverages you mentioned, even if slighltly different through the countries, are obtained through a distillation process, so that their appearence is somewhat pure and, at least visually, even more similar to the water.
Why the fire? Alcohol, once ingested produces heat (hence the fire), mostly at the centre of your chest (again, neat the heart, another universal symbol of life). Not to mention that it induces laughters, multiplies words, unties inhibitions. Let's say it intuitively reminds of life and movement as opposed to death and stagnation.

Historically distillated drinks were known among the Ancient Greeks and proabably even among Egyptians. But the word "Aqua Vite" is firstly used in the XIII century by a Franciscan monk, Bonaventura da Iseo in his work called Liber compostille. It was actually an alchemic compendium (it wasn't unusual for Franciscans to be interested in the most disparate branches of knowledge, including alchemy). He praises its virtues and explains how it can be useful in order to fight illness issues. Especially when Europe was infested with plague, it is said that distilled till the "quintessence", spirit was able to raise the dead (hence the "water of life").

Veritably the original, latin denomination is rather ambiguous: as vitae (life) and vitis (the grapevine, which is used to produce the spirit) sound so similar, and both make sense, we often find in these ancient, medical and alchemic books expressions that can be translated like "water of like" and "water of wine".

So my guess is that we can explain these similarities both from an anthropological and a philological point of view.