I have been unable to find any specific interviews or sources that explain this part of the movie, so the best I can do is present my own personal theory.
Small town conspiracy theorists
The movie is filmed and set in small town Pennsylvania where Shyamalan grew up. I believe this scene portrays an attitude of small town locals that he is used to from growing up. The irony being that typically you'd expect them to be sitting around watching the news and claiming its all an elaborate conspiracy to hide the existence of aliens, but when the news is actually about an alien invasion, instead of saying, "I knew it", they have hatched an entirely new conspiracy theory that they're just trying to sell sodas.
This does seem kind of strange specifically due to the 1984 commercial being shown in 2002. My theory on this again comes down to Shyamalan's personal experience. He was born in 1970 and probably would have remembered seeing this commercial when he was a teenager. This particular commercial adds to the humor of the moment too, at least in my opinion.
Easier to obtain rights?
Of course he would have seen all kinds of soda commercials in the 80s so this doesn't explain it in itself. Another possible reason for using Shasta specifically could be due to rights issues with showing a Coke or Pepsi commercial that Shasta might be more lax on due to them being more of a knock off cheap subsitute soda line.
Shasta TV Spots would likely replay old campaigns
One last thought is that I've seen commercials repeated on television these days that originated in the 80s. While not common, it is not impossible to imagine seeing a retrospective Shasta commercial on television. If Shasta were to purchase any TV spots then I would expect they would simply replay some of the marketing materials they created rather than spend a lot on a new and updated commercial.