Learn English – Why do some people prefix people’s names with “the”

It's a joke. Vodka is well known in the West as a Russian spirit, even as a stereotype of Russia. Skyy is saying that their vodka is so well filtered, they removed something that no one would have thought would be possible to remove from vodka (i.e., its association with Russia). That's all. If you start looking for alternate definitions of "Russia," or find it meaningful that Skyy is made by an Italian company, you're overthinking it.

"The Russia" is not normally acceptable, as you note, but it follows a widespread pattern of pairing "the" with unexpected words in informal conversation for effect, as a form of metonymy: wipe the smug right off his face.

The death penalty is referred to with an article because it is a specific thing—a penalty. It refers to something written in law, rather than all instances of it. There are multiple instances of the death penalty, referred to as death sentences.

In the linked article, another term for the death penalty is capital punishment. This does not use the definite article because punishment is a general idea—a term for the imposition of a penalty. We could, of course, say the punishment but that would refer to a specific instance of it.

Similarly, when talking about abortion, it is the general idea we are talking about—not something written somewhere in law books. There is no specific abortion we are talking about, no specific thing we can refer to as the abortion, unless we talk about an instance of it.

Here's a web page about when to not use the in English. Similar general things that do not need articles include

I like listening to music.

I watch sports.

Whereas, when we refer to a particular instance of these things:

I like listening to the music you sent me.

I watched the Bears game.