I know those points are tied to ranked matches as you cannot earn then by playing player matches only. Still, sometimes I will earn between 4-6 when defeating opponents and lose between 0-2 in the opposite case. Is there any word on how the system works specifically? Also, what does it even stand for?
PSR stands for Personal Skill Rating. Pretty similar to other fighting games, it's simply a value attached to your performance in ranked matches. It serves the purpose of giving players a quick sense at how good or how much another player has played, and it's likely used as a metric for pairing similarly-skilled players in ranked matchmaking. It's kind of hard to attach specific values to winning and losing though, since the amount of point you gain are dynamic and end up slowing greatly once you get into the range of about 85-115 wins. You stand to lose points when you lose a match so your PSR can be adjusted to a level where you'd be more likely to get matched up with someone at a similar skill level to you, and the opposite is true for winning.
It's a good, quick, reference for telling which players are at the top of the ladder and giving you a sense of your odds for winning a match; e.g. if another player has a couple hundred PSR more than you, then it's likely they've put more time in and/or are better than you.