There are various explanations here but, the most concise answer is: because it is a trope. And a well known trope at that.
Most of the answers to why it is seen are valid reasons that people in the past had for keeping their liquor in the bag, most notably the "legality" & "social acceptability" reasons. However in modern times it usually has no actual necessity, and is rarely done by real people.
Some people in real life might actually do this, usually as sort of tradition -- it's what they've seen the older generations do & they've never questioned why, just copied the same behavior, & whatever practical reasons started the behavior have been forgotten over generations of copying, i.e. "We do it this way because that's how it's always been done, because that's the way we've always done it"
Some might do it out of a sense of shame about drinking in public, of course it isn't so shameful to them that they'd actually refrain from drinking in public, just enough that they wish to "disguise" it.
Of course using the trope creates many conveniences for film/tv makers, and so they keep using it, even though it has little reflection in modern real life.
As to why it is a paper bag & not plastic, that is quite simple: when the practice of keeping liquor hidden in the bag was actually commonplace, plastic bags had not yet been invented, paper bags were the only option at stores, including liquor stores.
This is the very reason it was practiced, keeping the bottle in the bag provided "beverage anonymity" in public and was simple way to avoid a number of possible troubles for the drinker.
The practice backfired as it became more common, as eventually people began to assume that someone drinking from the bag had liquor; and thus the trope was born.